3. mathematics ii-matrices
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mathematics 2TRANSCRIPT
MATHEMATICS IIPAPER MATRICES
DISUSUN OLEH :
ARISA RISKI (D10011208)
PUGUH MERDHIYANTO (D100112014)
YUSRON ABDULLATIF RABBANI (D100110029)
CIVIL DEPARTMENT ENGINEERING FACULTY
MUHAMMADIYAH UNIVERSITY OF SURAKARTA
2012
MATRICES
THE ALGEBRA OF MATRICES
A matrix is a rectangular array of numbers. The numbers may be real or complex. It
may be represented as
A = ¿ [a11 a12⋯a1n ¿ ] [a21a22⋯a2 n ¿ ] [⋮¿ ]¿¿
¿
or asA = (aij )m x n
A matrix with m rows and n coloumns is called as m x n matrix and the size
(or dimension) of this matrix is said be m x n.
Two matrices are said to be equal provided they are of the same size and
corresponding elements are equal. For example
[abc ¿ ]¿¿
¿¿
if and only if
a = -1, b = 2, c = 5, d = 7, e = 3 and f = 1.
Definitions A matrix A = (aij)m x n is said to be a
(i) square matrix if m = n
(ii) row matrix if m = 1
(iii) coloumn matrix if n = 1
(iv) null or zero matrix if aij = 0 i and j
(v) diagonal matrix if m = n and aij = 0 i ≠ j
(vi) Scalar matrix if m = n and aij = 0 i ≠ j and aij = i
(vii) Unit or identity matrix if m = n and aij = 0 i ≠ j and aij = 1 i
(viii) Upper (Lower) triangular matrix if m = n and aij = 0 i > j
(aij = 0 i < j)
A matrix is said to be triangular if it is either lower or upper triangular matrix.
Addition of Matrix. Two matrices A and B can be added if and only they are of the
same size. For instance
[abc ¿ ]¿¿
¿¿
= ¿ [a + pb + qc + r ¿ ]¿¿
¿
Addition is not defined for matrices of different sized.
The additive inverse of a matrix A, denoted by –A, is the matrix whose
elements are the negatives of the corresponding elements of A. for example,
− ¿ [ab ¿ ] [cd ¿ ]¿¿
¿If A and B are two matrices of the same size, then the differences between A
and B is defined by
A – B = A + (-B)
Thus, the substraction is carried out term-by-term. For instance
[abc ¿ ]¿¿
¿¿
Properties of Addition If A, B and C are three matrices of the same size, then
A + B = B + A [Commutative law]
(A + B) + C = A + (B + C) [Associative law]
A + O = O + A [Additive property of zero]
A + (-A) = O
Where O is the null or zero matrix of the same size as that of A.
Scalar Multiplication. If A is a matrix and α is a scalar, then αA is defined as the
matrix obtained by multyplying every element of A by α. For example
3 ¿ [1−25 ¿ ]¿¿
¿
Properties of Scalar Multiplication If A, B are two matrices of the same size, and α,
β are two scalars, then
(α + β)A = αA + βA
(αβ)A = α (βA)
α (A + B) = αA + αB
Matrix Multiplication
Let A = (aij)m x n and B = (bij) r x s be two matrices. We stay that A and B are
comparable for the product AB if n = r, that is, if the number of coloumns of A is
equal to the number of rows of B.
Definition: Let A = (aij) m x n and B = (bij) m x n be two matrices. Their product AB is
the matrix C = (cij) m x n such that cij = ai1 b1j + ai2 b2j + ai3 b3j + ... + ain bnj for l ≤ i ≤
m, 1 ≤ j ≤ p. note that cij, the element of AB, has been obtained by multiplying ith
row of A, namely (ai1 ai2 ai3 ... ain)
With the jth column of B, namely
[b1 j¿ ] [b2 j ¿ ] [b3 j ¿ ] [. ¿ ] [. ¿ ] [. ¿ ] ¿¿
¿¿ = ( b1, b2j ... bnj)
Where A denotes the transpose of matrix A.
Properties of Matrix Multiplication
If A = (aij) m x n, B = (bij) n x p and C = (cij) p x q then
1. (AB) C = A (BC) [Associative law]
2. AIn = ImA = A
3. AB may not be equal to BA
4. k(AB) = (kA)B = A (kB) where k is a scalar.
5. If A is a square matrix, then
Am An = Am+n m, n N
(Am)n = Amn m, n N.
6. If A is an invertible matrix then
(A-1 B A)m = A-1 Bm A.
and A-m = (A-1)m m N.
TRANSPOSE OF A MATRIX
Definition: Let A = (aij)m x n be a matrix. The transpose of A, denoted by A’ or by A’
is the matrix A’ = (bij)n x m where bij = aji i and j.
By A we mean a matrix B = (bij) m x n where bij = a ij where a denotes conjugate of
a and by A* we mean.
A¿ = (A ) ' = ( A ' )Properties of Transpose of Matrix
1. (A + B)’ = A’ + B’
2. (kA)’ = kA’ where k is a scalar.
3. (AB)’ = B’ A’ [Reversal law]
4. If A is an invertible matrix, then (A-1)’ = (A’)-1
ADJOINT AND INVERSE OF A MATRIX
Let A = (aij)n x n be square matrix. The adjoint of A is defined to be the matrix adj. A
= (bij) n x n where
bij = Aji
where Aji is the cofactor of (j, i)th element of A.
Properties of Adjoint
1. A (adj A) = (adj A) A = |A| I n
2. adj(kA) = kn-1 (adj A)
3. adj(AB) = (adj B) (ajd A)
Definiton: A square matrix A is said to be singular if |A| = 0 and non-singular if
|A| ≠ 0.
Definition: Inverse of a square matrix A = (aij)n x n is the matrix B = (bij)n x n such that
AB = BA = In.
Infact A−1 = 1
|A|( adj . A )
Properties of Inverse
1. inverse of a matrix if it exists is unique.
2. AA-1 = A-1 A = In
3. (A-1)-1 = A
4. (kA)-+ = k-1 A-1 if k ≠ 0.
5. (AB)-1 = B-1 A-1 [reversal law]
6. For a matrix A = ¿ (ab ¿ )¿
¿¿
adj A = ¿ (d−b ¿ ) ¿
¿¿
and A−1 = 1
ad − bc¿ (d−b ¿ ) ¿
¿¿ if ad - bc ≠ 0
7. If A is a triangular matrix, then A-1 if it exists is a triangular matrix of the same
kind.
Infact if A = ¿ (a11 00¿ ) (a21a22 0 ¿)¿¿
¿ and a11a22a33 ≠ 0, then
A−1 = 1a11 a22 a33
¿ (a22 a33 00 ¿) (−a21 a33 a33 a11 0 ¿)¿¿
¿
Where A13 = cofactor of (1, 3)th element in i.e. A13 = ¿
|a21a22 ¿|¿¿
¿
8. If A = diag (1, 2, ..., n) then A-1 exists if i ≠ 0 i and
A-1 = diag (1-1, 2
-1, ..., n-1)
Also, Am = diag ( λm , λ2
m .. . λnm ) if m N.
9. If a square matrix A satisfies the equation a0 + a1x + a2 x+2+ + ... ar xr = 0,
then A is invertible if a0 ≠ 0 and its inverse is given by
A−1 = 1a0
[a1 I + a2 A + .. . + ar A r−1 ]
SOME DEFINITIONS AND RESULT
A square matrix is said to be symmetric if A’ = A and skew symmetric if A’ = - A.
A square matrix A is Hermitian if A* = A and skew Hermitian if A* = -A.
Result
1. The main diagonal elements of a skew symmetric matrix are zeros, i.e.a:: = 0
i
2. Determinant of a skew-symmetric matrix of odd order is zero and determinant
of a skew-symmetric matrix of even order is a perfect square.
3. Every square matrix can be written uniquely as a sum of symmetric and a skew
symmetric matrix, i.e. if A is a square matrix, then there exists a symmetric
matrix P and a skew symmetric matrix Q such that
Infact
P = 12
(A + A ' ) and Q = 12
( A − A ' )
4. For every square matrix A, matrices A + A’, AA’ and A’A are symmetric and
matrices A – A’ and A’ – A are skew symmetric.
A square matrix is said to be orthogona if AA’ = A’A = I.
A is said to be unitary if A*+A = I.
Result 1. If A is an orthogonal matrix, then |A| ≠ 0. Infact |A| = ± 1. And A , A’,
A* are also orthogonal.
2. If A and B are two orthogonal matrices, then AB and BA are both
orthogonal matrices.
A square matrix A is said to be a nilpotent matrix if there exist a positive integer m
such that Am = O.
The least positive integer m satisfying the condition Am = O is called the index of
the nilpotent matrix.
A square matrix A is said to be idempotent matrix if A2 = A and is said to be
involutory if A2+ = I.
If A = (aij) n x n is a square matrix, the trace of A denoted by tr (A) is sum of all the
main diagonal elements, i.e.
tr (A) = a11 + a22 + ... ann
RANK OF A MATRIX
If A = (aij) m x n is a matrix, and B is its submatrix of order, r, then |B|, the
determinant is called as r-rowed minor of A.
Definiton Let A = (aij) m x n be a matrix. A positive integer r is said to be rank of A if
(i) A possesses at least one r-rowed minor which is different from zero; and
(ii) Every (r + 1) rowed minor of A is zero.
From (ii), it automatically follows that all minors of higher order are zeros.
Result The rank of a matrix does not change when the following elementary row
operations are applied to the matrix.
(a) Two rows are interchanger (Ri Rj)
(b) A row is multiplied by a non-zero constant, (Ri kRi, with k ≠ 0)
(c) a constant multiple of another row is added to a given row (Ri Ri + kRj)
where i ≠ j.
note: The arrow means “replaced by”.
Note that the application of these elementary row operations does not change a
singular matrix to a non-singular matrix or a non-singular matrix to a singular
matrix. Therefore, the order of the largest non-singular square submatrix is not
affected by application of any of the elementary row operations. Thus, the rank of a
matrix does not change by application of any the elementary row operations.
A matrix obtained from a given matrix by applying any of the elementary row
operations is said to be equivalent to it. If A and B are two equivalent matrices, we
write A ~ B.
Note that if A ~ B, then (A) = (B).
By using the elementary row operations, we shall try to transform the given matrix
in the following form
(1∗¿∗¿ ) (01∗¿ ¿ ) (001∗¿ ) ( . .. .¿ ) ( . .. .¿ ) ( . . .. ¿ ) ¿¿
¿¿where * stands for zero or non-zero element. That is, we shall try to make aii as 1
and all the elements below aii as zero.
We illustrate the above procedure by the following illustration.
Illustration: Find the rank of the matrix
A = ¿ (2−31¿ ) (357 ¿ ) ¿¿
¿Solution
Step 1: As a first step we must get a 1 in the first column of A. For this we
substract Row 1 from the Row 2.
(2−31 ¿ ) (357 ¿ ) ¿¿
¿¿
Step 2: We must get a 1 in the upper left corner. For this we interchange Row 1 and
Row 2.
(2−31 ¿ ) (186 ¿ ) ¿¿
¿¿Step 3: We must get zeros at the two remaining two places in the first column. For
this we multiply R1 by – 2 and add it to R2 and multiply R1 by – 5 and add it to R3.
(186 ¿ ) (2−31 ¿ ) ¿¿
¿¿
(186 ¿ ) (0−19−11 ¿ ) ¿¿
¿¿Step 4: We must have 1 in the second column. This 1 should not be in the first
column. Also, you should not be tempted to use R1 to obtain this 1. For if we try to
use R1. Then two zeroes obtained in the first column will be destroyed. We multiply
R2 by – 2 and add it to R3.
0 38 22 - R2
(186 ¿ ) (0−19−11 ¿ ) ¿¿
¿¿Step 5: We now obtain a 1 at the (2, 2)th place. For this we interchange R2 and R3.
(186 ¿ ) (0−19−11 ¿ ) ¿¿
¿¿Step 6: We must get a zero at (3, 2)th place. For this we multiply R2 by 19 and add
it to R3.
(186 ¿ ) (010 ¿ ) ¿¿
¿¿
This matrix is in the desired triangular form. Recall if A ~ B then (A) = (B).
Thus, rank of the given matrix A is equal to the rank of the matrix.
B = ¿ (186 ¿ ) (010¿ )¿¿
¿Hance, (A) = 3.
Remark
After obtaining 1 at (1, 1)th place and zeros at the remaining places in the first
column, forget the first row. Do not use the first row. Do not use the first row to
manipulate elements in the second or any other column. If you try to do so the
zeros in the first column will be destroyed.
After obtaining 1 at (2, 2)th place and zeros at the remaining places in the
second column, forget the second row. Do not use it for manipulating elements
in the remaining columns. The same remark applies to the remaining columns.
If A is a non-regular matrix of order n x n, then e (A) = n.
SYSTEM OF LINEAR EQUATIONS
Let us consider the following m linear equations in n unknowns:
a11 x1 + a12 x2 + ... + a1n xn = b1
a21 x1 + a22 x2 + ... + a2n xn = b2
.
.
.
am1 x1 + am2 x2 + ... + amn xn = bm
where b1, b2, ... bm are not all zero.
The m x n matrix
(a11 a12…a1n ¿ )(a21 a22⋯a2n ¿ ) (. . .¿ ) ( . .. .¿ ) ( . .. ¿ ) ¿¿
¿¿ is called the coefficient matrix
of the system of linear equations. Using it, we can now write these equations as
follows:
(a11a12⋯a1n ¿ )(a21 a22⋯a2n ¿ ) (. . .¿ ) ( . .. .¿ ) ( . .. ¿ ) ¿¿
¿¿We can abbreviate the above matrix equation to AX = B, where
A = ¿ (a11 a12⋯a1n ¿ ) (a21a22⋯a2n ¿ ) ( .. .¿ ) ( . .. . ¿ ) (. . .¿ )¿¿
¿and
X = ¿ ( x1 ¿ )( x2¿ ) ( .¿ ) ( . ¿ ) (. ¿ ) ¿¿
¿By a solution of (1) we mean a set of values x1, x2, ..., xn such that (1) reduces to an
identity.
The augmented matrix for system of equations.
AX = B
is the matrix (A/B). This matrix is obtained by adding (n + 1)th column to A. the
elements of column are the constants b1, ..., bm.
Result The system of equations
a11 x1 + a12 x2 + … a1n xn = b1
a21 x1 + a22 x2 + … a2n xn = b2 …(1)……………………………………
am1 x1 + am2 x2 + … amn xn = bn
is consistent (that is, possesses a solution) if and only if the coefficient matrix.
A = ¿ (a11 a12⋯a1n ¿ ) (a21a22⋯a2n ¿ ) (⋯⋯⋯⋯¿ ) ¿¿
¿and the augmented matrix
(A | B) =
(a11 a12⋯a1n | b1 ¿) (a21a22⋯a2 n | b2 ¿) (⋯⋯⋯⋯ | ⋯¿ ) ¿¿
¿¿have the same rank.
We split the remaining result in two cases.
Case 1. If the system of equations in (1) is consistent and m ≥ n, then
(i) if (A) = (A | B) = n, then the system of equations has a unique solution,
(ii) if (A) = (A | B) = r < n, then the (n – r) unknowns are assigned
arbitrary values and the remaining r unknowns can be found in terms of those (n –
r) unknowns which have already been assigned values.
Case 2. If the system of equations in (1) is consistent and m < n, then
(i) if (A) = (A | B) = m, then (n – m) unknowns can be assigned arbitrary
values and the values of the remaining m unknowns can be found in terms of those
(n – m) unknowns which have already been assigned values.
(ii) if (A) = (A | B) = r < m then (n – r) unknowns can be assigned arbitrary
values and the values of the remaining r unknowns can be found in terms of those
(n – r) unknowns which have already been assigned values.
Finding Inverse by Elementary Row Operations
To find inverse of a square matrix A we begin with be augmented matrix [A | In]. If
a sequence of elementary row operations transforms this matrix to [In | B], then B is
A-1. However, if at any step we obtain all zeros in a row on the left of the vertical
line, the matrix A is not invertible.
SOLUTION OF A SYSTEM OF EQUATION AX = B
Unique Solution
The system of equation
AX = B
has a unique solution if |A| ≠ 0 and it is given by X = A-1 B
Infinite Number of Solution
If |A| = 0 , and (Adj A)B = 0, the system has infinite number of solutions.
No Solution
If |A| = 0 and (Adj A)B ≠ 0, the system of equations has no solution.
SOLUTION OF A SYSTEM OF HOMOGENEOUS LINEAR EQUATIONS
AX = 0
The system
AX = 0
has a unique solution if |A| ≠ 0 and it is the trivial solution viz.
x1 = x2 = … = xn = 0
If |A| = 0 , the system has infinite number of solutions.
Also if AX = 0 has at least one non-zero solutions, then |A| = 0 .
The following Tree diagram is helpful.
Solved Examples
Example 1
If
[14 ¿ ] ¿¿
¿¿, y < 0 then x – y + z is equal to
(a) 5 (b) 2 (c) 1 (d) -3
Ans (a)
Solution By the equality of two matrices, x = 1, y2 = 4, z = 2
x = 1, y = -2, z = 2 as y < 0.
x – y + z = 1 + 2 + 2 = 5
Example 2
If
A = [1−23 ] , B = ¿ [2 ¿ ] [−3 ¿ ]¿¿
¿, then AB is equal to a
System of equations
is consistent
System of equations
is consistent
Infinite number
of solution
Trival solution
(A) = (A |
B)
(A) = (A |
B)|A| = 0|A| ≠ 0
B = 0 B ≠ 0
AX =
B
(a)
[2¿ ] [−3 ¿ ]¿¿
¿¿(b)
[2¿ ] [6 ¿ ] ¿¿
¿¿
(c) [26−3 ] (d) none of these
Ans (d)
Solution
A = [1−23 ] , B = ¿ [2 ¿ ] [−3 ¿ ]¿¿
¿
Example 3
If A = ¿ [−i 0¿ ]¿
¿¿, then A’ A is equal to
(a) I (b) –iA (c) – I (d) iA
Ans (c)
Solution We have
A ' A = ¿ [−i0 ¿ ]¿¿
¿
Example 4
If Aα = ¿
[cos α sin α ¿ ]¿¿
¿, then Aα Aβ is equal to
(a) Aα + Aβ (b) Aα
β
(c) Aα− A β (d) none of these
Solution We have
Aα A β = ¿[cos α sin α ¿ ]¿
¿¿
= ¿ [cos α cos β −sin α sin β cos α sin β + sin α cos β ¿ ]¿¿
¿
= ¿ [cos (α + β ) sin (α + β ) ¿ ]¿¿
¿
Example 5
Let A and B be two 2 x 2 matrices. Consider the statements.
(i) AB = O A = O or B = O
(ii) AB = I2 A = B-1
(iii) (A + B)2 = A2 + 2AB + B2
Then
(a) (i) is false, (ii) and (iii) are true
(b) (i) and (iii) are false, (ii) is true
(c) (i) and (ii) are false, (iii) is true
(d) (ii) and (iii) are false, (i) is true
Ans (b)
Solution (i) is false.
IfA = ¿ [01¿ ]¿
¿¿
then
AB= ¿ [00 ¿ ]¿¿
¿
Thus, AB = 0 A = O or B = O
(iii) is false since matrix multiplication is not communicative.
(ii) is true as product AB is an identity matrix, if B is inverse of the matrix A.
Example 6
If A − 2B = ¿ [15 ¿ ]¿
¿¿ and
2 A − 3B = ¿ [−25 ¿ ]¿¿
¿, then matrix B is equal to
(a)
[−4−5 ¿ ]¿¿
¿¿(b)
[06 ¿ ] ¿¿
¿¿(c)
[2−1 ¿ ] ¿¿
¿¿(d)
[6−1¿ ]¿¿
¿¿
Solution We have
B = (2 A − 3 B ) − 2 (A − 2 B ) = 2 ¿ [−4−5 ¿ ]¿¿
¿
Example 7
If A and B two are 3 x 3 matrices, then which one of the following is not true:
(a) (A + B)’ = A’ + B’ (b) (AB)’ = A’ B’
(c) det (AB) = det (A) det (B) (d) A (adj A) = |A| I 3
Ans (b)
Solution If A and B are two 3 x 3 matrices, then
(AB)’ = B’A’ [Reversal Law]
and not (AB)’ = A’B’.
Example 8
If A = ¿ [cos φ−sin φ ¿ ]¿
¿¿, then
(a) A is an orthogonal matrix (b) A is a symmetric matrix
(c) A is a skew symmetric matrix (d) none of these
Ans (a)
Solution We have
= ¿ [cos φ−sin φ ¿ ]¿¿
¿
= ¿ [cos2 φ + sin2 φ cos φ sin φ − sin φ cos φ ¿ ]¿¿
¿
= ¿ [10 ¿ ]¿¿
¿
Similarly, A’ A = I2
Thus, A is an orthogonal matrix.
Example 9
Matrix [12 ] ¿¿
is equal to
(a) [122 ] (b) [23 ]
(c) [22 ] (d) none of these
Ans (c)
Solution We have
[12 ] ¿¿= [12 ] ¿ [−2 + 10 ¿ ]¿
¿¿
= [8 + 14 ] = [22 ]
Example 10
If A = ¿ [5−13 ¿ ]¿
¿¿
, then (AB)’ is equal to
(a)
[78 ¿ ]¿¿
¿¿(b)
[−78 ¿ ]¿¿
¿¿
(c)
[78 ¿ ]¿¿
¿¿(d) none of these
Ans (c)
Solution A = - A’
A is a skew symmetric matrix
diagonal elements are zeros
x = 0, y = 0
x + y = 0
Example 12
If
A = ¿ [a2abac ¿ ] [ abb2bc ¿ ]¿¿
¿ then the product AB is
equal to
(a) O (b) A (c) B (d) I
Solution We have
AB = ¿ [0− abc + abc a2 c + 0− a2 c−a2 b + a2 b + 0 ¿ ] [0 − b2 c + b2 cabc + 0− abc−ab2 + ab2 + 0 ¿ ]¿¿
¿
Example 13
If A is an orthogonal matrix, then A-1 equals
(a) A (b) A’
(c) AA’ (d) none of these
Ans (b)
Solution By definition, A square matrix A is said to be orthogonal if
AA’ = A’A = I
A-1 = A’
Example 14
If A is an invertible matrix and B is an orthogonal matrix, of the order same as that
of A, then C = A-1 BA is
(a) an orthogonal matrix (b) symmetric matrix
(c) skew symmetric matrix (d) none of these
Ans (d)
Solution
Let B = ¿ [cos (π / 2 ) sin (π / 2 ) ¿ ] ¿
¿¿
and A = ¿ [13 ¿ ]¿
¿¿
Note that B is an orthogonal matrix.
C = A−1 BA = ¿ [1−3 ¿ ]¿¿
¿
= ¿ [3 10¿ ]¿¿
¿
Note that C is neither symmetric, nor skew symmetric and nor orthogonal.
Example 15
Let E (α ) = ¿ [cos2 α cos α sin α ¿ ]¿
¿¿
if and differs by an odd multiply of
/2, then E () E () is a
(a) null matrix (b) unit matrix
(c) diagonal matrix (d) orthogonal matrix
Ans (a)
Solution We have
E (α ) = ¿ [cos2 α cos α sin α ¿ ]¿¿
¿
= ¿ [cos α cos β cos (α − β ) cos α sin β cos (α − β ) ¿ ] ¿¿
¿
As and differ by an odd multiple of /2, - = (2n +1) /2 for some integer n.
Thus, cos [ (2 n + 1 ) π /2 ] = 0
E () E () = O
Example 16
If
[21¿ ]¿¿
¿¿ then matrix A equals
(a)
[75 ¿ ]¿¿
¿¿(b)
[21¿ ]¿¿
¿¿(c)
[71¿ ]¿¿
¿¿(d)
[53 ¿ ]¿¿
¿¿
Ans (a)
Solution
If XAY = I, then A = X-1 Y-1 = (YX)-1
In this caseYX = ¿ [−32¿ ]¿
¿¿
A = ¿ [85 ¿ ]¿
¿¿
Example 17
The matrix A satisfying
A ¿ [15 ¿ ]¿¿
¿ is
(a)
[32¿ ]¿¿
¿¿(b)
[3−16 ¿ ]¿¿
¿¿(c)
[3−16 ¿ ]¿¿
¿¿(d)
[3−3 ¿ ]¿¿
¿¿
Ans (b)
Solution We know that if AC = B, then A = BC-1
A = ¿ [3−1¿ ]¿
¿¿
= ¿ [3−1 ¿ ] ¿¿
¿
Example 18
If product of matrix A with
[11 ¿ ]¿¿
¿¿ is
[32¿ ]¿¿
¿¿, then A-1 is given by
(a)
[0−1¿ ]¿¿
¿¿(b)
[0−1¿ ]¿¿
¿¿
(c)
[01¿ ]¿¿
¿¿(d) none of these
Ans (c)
Solution If AB = C, then B-1 A-1 = C-1
A-1 = BC-1
HereA ¿ [11 ¿ ]¿
¿¿
A−1= ¿ [11¿ ]¿
¿¿
= ¿ [11 ¿ ]¿¿
¿
Example 19
If A and B are two skew symmetric matrices of order n, then
(a) AB is a skew symmetric matrix
(b) AB is a symmetric matrix
(c) AB is a symmetric matrix if A and B commute
(d) none of these
Ans (c)
Solution We are given
A’ = -A and B’ = - B
Now, (AB)’ = B’ A’ = (-B) (-A) = BA
= AB if A and B commute
Example 20
Which of the following statements is false:
(a) If |A| = 0 , then |adj A|= 0
(b) Adjoint of a diagonal matrix of order 3 x 3 is a diagonal matrix
(c) Product of two upper triangular matrices is a upper triangular matrix
(d) adj (AB) = adj (A) adj (B)
Ans (d)
Solution We have
adj (AB) = adj (B) adj (A)
and not adj (AB) = adj (A) adj (B)
Example 21
If A and B are symmetric matrices, then AB – BA is a
(a) symmetric matrix (b) skew symmetric matrix
(c) diagonal matrix (d) null matrix
Ans (b)
Solution We are given A’ = A, B’ = B
Now (AB – BA)’ = (AB)’ – (BA)’
= B’ A’ – A’ B’
= BA - AB
= - (AB – BA)
i.e. (AB – BA)’ = - (AB – BA)
Hence AB – BA is a skew symmetric matrix.
Example 22
If D = diag (d1, d2, …, dn) where d1 ≠ 0, for I = 1, 2, …, n, then D-1 is equal to
(a) D (b) 2D
(c) diag (d1−1 , d2
−1 , .. . , dn−1) (d) Adj D
Ans (c)
Solution See Theory
Example 23
The inverse of a symmetric matrix (if it exists) is
(a) a symmetric matrix (b) a skew symmetric matrix
(c) a diagonal matrix (d) none of these
Ans (a)
Solution Let A be an invertible a symmetric matrix.
We have AA-1 = A-1 A = In
(AA-1)’ = (A-1 A)’ = (In)
(A-1)’ A’ = A’ (A-1)’ = In
(A-1)’ A= A (A-1)’ = In
(A-1)’ = A-1 [inverse of a matrix is unique]
Example 24
The inverse of a skew symmetric matrix (if it exists) is
(a) a symmetric matrix (b) a skew symmetric matrix
(c) a diagonal matrix (d) none of these
Ans (b)
Solution We have A’ = - A
Now AA-1 = A-1 A = In
(AA-1)’ = (A-1 A)’ = (In)’
(A-1)’ A’ = A’ (A-1)’ = In
(A-1)’ (-A) = (-A) (A-1)’ = In
Thus, (A-1)’ = - (A-1) [inverse of a matrix is unique]
Example 25
The inverse of a skew symmetric matrix of odd order is
(a) a symmetric matrix (b) a skew symmetric matrix
(c) diagonal matrix (d) does not exist
Ans (d)
Solution Let A be a skew symmetric, matrix of order n. By definition
A’ = -A
|A '| = |− A|
|A| = (- 1)n |A|
|A| = - |A| [∵ n is odd ]
2 |A| = 0
|A| = 0
A-1 does not exist.
Example 26
If A is an orthogonal matrix, then |A| is(a) 1 (b) -1 (c) ± 1 (d) 0
Ans (c)
Solution As A is an orthogonal matrix,
A’ A = AA’ = In
|A ' A| = |A A '| = |I n|
|A '| |A|= 1
|A| |A| = 1
|A|2 = 1 |A| = ± 1
Example 27
If
A = ¿ [102¿ ] [51 x ¿ ] ¿¿
¿ is a singular matrix, then x is equal to
(a) 3 (b) 5 (c) 9 (d) 11
Ans (c)
Solution As A is a singular matrix
|A| = 0
|100 ¿||51 x − 10 ¿|¿¿
¿¿[using C3 C3 – 2 C1]
|1 x − 10 ¿|¿¿
¿¿= 0 -1 – x + 10 = 0
x = 9.
Example 28
The value of x for which the matrix
A = ¿ [2/ x−12 ¿ ] [1 x2 x2 ¿ ] ¿¿
¿is singular is
(a) ± 1 (b) ± 2
(c) ± 3 (d) none of these
Ans (a)
Solution We have
|A|= ( 2x ) ¿|x2 x2¿|¿
¿¿
= 2x
(0 ) + 2 − 2 x2 + 2 ( 1x − x)=
2 x (1 − x2) + 2 (1 − x2)x
=2 ( x + 1 )2 (1 − x )
x
Now, |A| = 0 x = ± 1
Example 29
If square matrix A is such that 3A2 + 2A2 + 5A + I = O, then A-1 is equal to
(a) 3A2 + 2A + 5I (b) – (3A2 + 2A + 5I)
(c) 3A2 – 2A – 5I (d) none of these
Solution We have
A (3A2 + 2A + 5I) = - I
A-1 = - (3A2 + 2A + 5I) [ Inverse of a matrix is unique]
Example 30
If A is a square matrix such that A2 + I = 0, then A equals
(a)
[10 ¿ ]¿¿
¿¿(b)
[ i0 ¿ ]¿¿
¿¿(c)
[12¿ ]¿¿
¿¿(d)
[−10 ¿ ]¿¿
¿¿
Ans (b)
Solution We have
[ i0 ¿ ]¿¿
¿¿
[ i0 ¿ ]¿¿
¿¿ =
[−10 ¿ ]¿¿
¿¿
Example 31
Let A, B, C be three square matrices of the same order, such that whenever AB =
AC then B = C, if A is
(a) singular (b) non-singular
(c) symmetric (d) skew-symmetric
Ans (b)
Solution If A is non-singular, A-1 exists.
Thus, AB = AC A-1 (AB) = A-1 (AC)
(A-1 A) B = (A-1 A) C IC
B = C
Example 32
If the product of the matrix
B = ¿ [264 ¿ ] [101 ¿ ]¿¿
¿ with a matrix A has inverse
C = ¿ [−101 ¿ ] [113 ¿ ]¿¿
¿, then A-1 equals
(a)
[−3−55 ¿ ] [09 14 ¿ ]¿¿
¿¿(b)
[−355 ¿ ] [009 ¿ ]¿¿
¿¿
(c)
[−3−5−5 ¿ ] [002 ¿ ] ¿¿
¿¿(d)
[−3−3−5 ¿ ] [092 ¿ ]¿¿
¿¿Ans (c)
Solution We have (BA)-1 C A-1 B-1 = C A-1 = CB
A−1 = ¿ [−101 ¿ ] [113 ¿ ]¿¿
¿
Example 33
If w is a complex cube root of unity, then the matrix
A = ¿ [1 w2w ¿ ] [w2w 1¿ ]¿¿
¿ is a
(a) singular matrix (b) non-singular matrix
(c) skew symmetric matrix (d) none of these
Ans (a)
Solution We have
|A|= ¿ [1w2 w ¿ ] [w2 w 1¿ ]¿¿
¿[using C1 C1 + C2 + C3]
= ¿|0 w2 w ¿||0w 1 ¿|¿¿
¿ A is a singular matrix.
Example 34
If
A = ¿ [012 ¿ ] [123 ¿ ] ¿¿
¿ and
A−1 = ¿ [1 /2−1/21/2¿ ] [−43 y ¿ ]¿¿
¿, then
(a) x = 1, y = -1 (b) x = -1, y = 1
(c) x = 2, y = - ½ (d) x = ½, y = ½
Ans (a)
Solution We have
[100 ¿ ] [010 ¿ ]¿¿
¿¿
= ¿ [10 y + 1 ¿ ] [012 ( y + 1) ¿ ] ¿¿
¿
1 – x = 0, x – 1 = 0, y + 1 = 0, y + 1 = 0, 2 + xy = 1
x = 1, y = -1
Example 35
If A = ¿ [01 ¿ ]¿
¿¿
then A4 is
(a)
[10 ¿ ]¿¿
¿¿(b)
[11 ¿ ]¿¿
¿¿
(c)
[00 ¿ ]¿¿
¿¿(d)
[01¿ ]¿¿
¿¿
Ans (a)
Solution We have
A2= ¿ [01¿ ]¿¿
¿
A4= A2 A2 = II = I
Example 36
If A = ¿ [3−4 ¿ ]¿
¿¿, then An (where n N) is
(a)
[3n−4n ¿ ] ¿¿
¿¿(b)
[n + 25 − n ¿ ] ¿¿
¿¿
(c)
[3n (−4 )n ¿ ] ¿¿
¿¿(d) none of these
Ans (d)
Solution We have
A2 = ¿ [3−4 ¿ ]¿¿
¿
For n = 2, none of (a), (b), (c) match with the actual answer.
Thus, answer is (d).
Example 37
If A and B are two matrices such that AB = B and BA = A, then A2 + B2 is equal to
(a) 2AB (b) 2BA
(c) A + B (d) AB
Solution We have
A2 + B2= (BA)2 + (AB)2
= (BA) (BA) + (AB) (AB)
= (B (AB) A + A (BA) B
= B (BA) + A (AB)
= BA + AB = A + B
Example 38
If
A = ¿ [201 ¿ ] [213 ¿ ] ¿¿
¿, then A2 – 5A + 6I is equal to
(a)
[1−1−3¿ ] [−1−1−10¿ ]¿¿
¿¿(b)
[11−5¿ ] [−1−14 ¿ ]¿¿
¿¿(c) 0 (d) I
Ans (a)
Solution
A2 = ¿ [201 ¿ ] [213 ¿ ]¿¿
¿Now, A2 – 5A + 6I
= ¿ [5−12¿ ] [9−25¿ ]¿¿
¿
= ¿ [5 − 10 + 6−1−+ 02− 5 + 0 ¿ ] [9− 10 + 0−2− 5 + 65 − 15 + 0 ¿ ]¿¿
¿
= ¿ [1−1−3 ¿ ] [−1−1−10 ¿ ] ¿¿
¿
Example 39
The inverse of the matrix
A = ¿ [100 ¿ ] [a 10 ¿ ] ¿¿
¿ is
(a)
[100 ¿ ] [−a 10 ¿ ]¿¿
¿¿(b)
[100 ¿ ] [−a 00 ¿ ]¿¿
¿¿
(c)
[100 ¿ ] [−a 00 ¿ ]¿¿
¿¿(d) none of these
Ans (a)
Solution Using the formula for inverse of a 3 x 3 triangular matrix given in
theory, A-1 is the matrix given in (a).
Example 40
If
A = ¿ [3−34 ¿ ] [2−34 ¿ ]¿¿
¿ then A-1 is
(a) A (b) A2 (c) A3 (d) A4
Ans (c)
Solution To show that A-1 = B AB = I. We have
A2= ¿ [3−34 ¿ ] [2−34 ¿ ]¿¿
¿
A ( A2) = ¿ [3−34 ¿ ] [2−34 ¿ ]¿¿
¿
= ¿ [1∗¿ ¿ ] [−8∗¿ ¿ ]¿¿
¿[need not evaluated the remaining terms as A3 ≠ I3]
Next
A4 = ( A2) ( A2 ) = ¿ [3−44 ¿ ] [0−10 ¿ ]¿¿
¿
= ¿ [100 ¿ ] [010¿ ]¿¿
¿Thus, A-1 = A3
EXERCISES
1. If I = ¿ [10 ¿ ] ¿
¿¿
, then B equals
(a) (cos ) I + (sin ) J
(b) (sin ) I + (cos ) J
(c) (cos ) I – (sin ) J
(d) – (cos ) I + (sin ) J
2. If a matrix A is both symmetric and skew-symmetric, then
(a) A is a diagonal matrix
(b) A is a null matrix
(c) A is a unit matrix
(d) A is a triangular matrix
3. If A is a skew symmetric of odd order, then |A| equals
(a) 0
(b) -1
(c) 1
(d) none of these
4. If
A = ¿ [50−1 ¿ ] [070 ¿ ]¿¿
¿ then
(a) A is a diagonal matrix
(b) A is symmetric
(c) A is skew symmetric
(d) A is an upper triangular
matrix
5. If A is a square matrix of order 3 such that A2 = 2A, then |A|2 is equal to
(a) 2 |A|
(b) 8 |A|(c) 16 |A|(d) 0
6. If A is a square matrix then which one of the following is not a symmetric
matrix
(a) A + A
(b) AA’
(c) A’ A
(d) A – A’
7. If A = (aij)3 x 3 where aij = i + j, then
(a) A is symmetric
(b) A is skew symmetric
(c) A is a triangular matrix
(d) A is a singular matrix
8. If A = (aij)3 x 3 is a matrix satisfying the equation x3 – 3x + 1 = 0, then
(a) A is a unit matix
(b) A is singular matrix
(c) A is non-singular matrix
(d) none of these
9. If A and B are two square matrices of the same size, then (A + B) 2 = A2 + 2AB
+ B2 can hold if and only if
(a) AB = BA
(b) AB + BA = O(c) |A B|≠ 0
(d) |A B|= 0
10. If
[ i0 ¿ ]¿¿
¿¿, then X is equal to
(a)
[0−1¿ ]¿¿
¿¿
(b)
[01¿ ]¿¿
¿¿
(c)
[10 ¿ ]¿¿
¿¿
(d) none of these
11. If A = ¿ [0−i ¿ ] ¿
¿¿
, then AB + BA is
(a) null matrix
(b) unit matrix
(c) invertible matrix
(d) none of these
12.
A = ¿ [123 ¿ ] [123 ¿ ]¿¿
¿, then A is a nilpotent matrix of index
(a) 2
(b) 3
(c) 4
(d) 5
13. If A is an unitary matrix, then |A| is equal to
(a) 1
(b) -1
(c) ± 1
(d) none of these
14. If A = 1
2¿ (−1− √3¿ )¿
¿¿, then A-1 – A2 is equal to a
(a) null matrix
(b) invertible matrix
(c) unit matrix
(d) none of these
15. If C is a 3 x 3 matrix satisfying the relation C2 + C = I, then C-2 is given by
(a) 2 C
(b) 3 C
(c) C
(d) none of these
16. If A, B and C are three square matrices of the same size such that B = CA C -1,
then CA3 C-1 is equal to
(a) B
(b) B2
(c) B3
(d) B9
17. If X is a 2 x 3 matrix such that |X ' X|≠ 0 , and A = I2 – X (X’ X)-1 X’ then A2 is
equal to
(a) A
(b) I
(c) A-1+
(d) none of these
18. The matrix A = ¿ ( p−q ¿ ) ¿
¿¿ is orthogonal if and only if
(a) p2 + q2 = 1
(b) p2 = q2
(c) p2 = q2 + 1
(d) none of these
19. The values of for which the matrix
A = ¿ ( λ 0 λ ¿ ) ( λ 0−λ ¿ ) ¿¿
¿ is orthogonal is
(a) ± 1
(b) ± 1/ √3
(c) ± ½
(d) none of these
20. The values of a for which the matrix
A = ¿ (aa2− 1−3 ¿) (a + 12 a2 + 4 ¿ )¿¿
¿ is symmetric
are
(a) -1
(b) -2
(c) 3
(d) none of these
21. Let
Ai = ¿(132 ¿ ) (25 t ¿ ) ¿
¿¿, then the value(s) of t for which inverse of At does
not exist.
(a) -2, 1
(b) 3, 2
(c) 2, -3
(d) 3, -1
22. If A = ¿ [a + ib c + id ¿ ]¿
¿¿
, where a2 + b2 + c2 +d2 = 1, then A-1 is equal to
(a)
[a − ib−c + id ¿ ]¿¿
¿¿
(b)
[a − ibc − id ¿ ]¿¿
¿¿
(c)
[a − ib−c − id ¿ ]¿¿
¿¿
(d) none of these
23. If
A = ¿ [ 12 (eix + e−ix ) 12
(eix− e−ix )¿ ]¿¿
¿ then A-1 exist
(a) for all real x
(b) for positive real x only
(c) for negative real x only
(d) none of these
24. If A = ¿ [abb2 ¿ ] ¿
¿¿
, then A2 is equal
(a) O
(b) I
(c) – I
(d) none of these
25. If A is 2 x 2 matrix such that A2 = O, then tr (A) is
(a) 1
(b) -1
(c) O
(d) none of these
26. If A = ¿ [ab ¿ ]¿
¿¿ such that A satisfies the relation A2 – (a + d) A = O, then
inverse of A is
(a) I
(b) A
(c) (a + d) A
(d) none of these
27. If A = ¿ [32¿ ]¿
¿¿
, then A-1 is
(a)
127¿ [1−26¿ ]¿
¿¿
(b)
127¿ [−1−26 ¿ ]¿
¿¿
(c)
127¿ [1−26¿ ]¿
¿¿
(d)
127¿ [1 26 ¿ ]¿
¿¿
28. If A is a skew Hermitian matrix, then the main diagonal elements of A are all
(a) real
(b) positive
(c) negative
(d) none of these
29. If
A = ¿ [121¿ ] [01−1 ¿ ] ¿¿
¿, then A3 – 3A2 – A – 9I is equal to
(a) O
(b) I
(c) A(d) A2
30. If
3 A = ¿ [122¿ ] [21−2 ¿ ]¿¿
¿ and AA’ = I, then x + y is equal to
(a) -3
(b) -2
(c) -1
(d) 0
31. If the system of equations ax + y = 3, x + 2y = 3, 3x + 4y = 7 is consistent, then
value of a is given by
(a) 2
(b) 1
(c) -1
(d) 0
32. If the system of equations x + 2y – 3z = 1, (p + 2) z = 3, (2p + 1) y + z = 2 is
inconsistent, then the value of p is
(a) -2
(b) – ½
(c) 0
(d) 2
33. The system of linear equations x + y + z = 2, 2x + y – z = 3, 3x + 2y + kz = 4
has a unique solution if
(a) k ≠ 0
(b) -1 < k < 1
(c) -2 < k < 2
(d) k = 0
34. If A= ¿ [4 x + 2 ¿ ]¿
¿¿ is an invertible matrix, then x cannot take value
(a) -1
(b) 2
(c) 3
(d) none of these
35. If A and B are two square matrices of the same order, then which of the
following is true
(a) (AB)’ = A’ B’
(b) (AB)’ = B’ A’(c) |AB|= 0 ⇒|A|= 0 and |B|= 0
(d) none of these
36. The values of for which the system of equations x + y + z = 1, ix + 2y + 4z =
, x + 4y + 10z = 2 is consistent, are given by
(a) 1, -2
(b) -1, 2
(c) 1, 2
(d) none of these
37. If x, y, z are in A.P with common difference d and the rank of the matrix
[45 x ¿ ] [56 y ¿ ]¿¿
¿¿ is 2, then value of d and k are
(a) x, 5
(b) x/2, 6
(c) arbitrary number, 7
(d) x/4, arbitrary number, 7
38. If A and B are two 3 x 3 matrices and |A|≠ 0 , then which of the following are
not true?
(a) |A B|= 0 ⇒ |B|= 0
(b) |A B|≠ 0 ⇒ |B|≠ 0(c) |A
−1|=|A|−1
(d) |A + A|= 2 |A|
39. If A = ¿ [ i−i ¿ ]¿
¿¿
, then A8 equals
(a) 64 B
(b) 32 B
(c) 16 B
(d) 8 B
40. If
A = ¿ (23 − i−i ¿ ) (3 + iπ 7 + i ¿ ) ¿¿
¿, then A is
(a) symmetric
(b) Hermitian
(c) skew Hermitian
(d) none of these
ANSWERS
1. (a)
2. (b)
3. (a)
4. (d)
5. (b)
6. (d)
7. (d)
8. (c)
9. (a)
10. (b)
11. (a)
12. (a)
13. (d)
14. (a)
15. (b)
16. (c)
17. (a)
18. (a)
19. (d)
20. (d)
21. (c)
22. (c)
23. (a)
24. (a)
25. (c)
26. (d)
27. (c)
28. (d)
29. (a)
30. (a)
31. (a)
32. (a)
33. (a)
34. (d)
35. (b)
36. (c)
37. (c)
38. (d)
39. (a)
40. (b)
Source :
TATA McGRAW-HILL`S COMPANIES,2005-2006,Complite Mathematics For AIEEE (ALL
INDIAN ENGINEERING ENTRANCE EXAMINATION).