matrices

Presented by

Ajay Gupta

AJAY GUPTA PGT MATHSCONT. NO. 9868423152

KV VIKASPURI, NEW DELHI

MATRICESMATRICES A matrix is a rectangular array A matrix is a rectangular array

(arrangement) of numbers real (arrangement) of numbers real or imaginary or functions kept or imaginary or functions kept inside braces () or [ ]subject to inside braces () or [ ]subject to certain rules of operations.certain rules of operations.

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ORDER OF A MATRIXORDER OF A MATRIX

A matrix having ‘m’ number of A matrix having ‘m’ number of rows rows and ‘n’ number of and ‘n’ number of columnscolumns is said to be is said to be of order ‘m n’of order ‘m n’

I rowI row

II rowII row

III rowIII row

I II IIII II III

ColumnsColumns

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Notation of a MatrixNotation of a Matrix

1.1. In compact form matrix is represented by In compact form matrix is represented by A = [a A = [a i j i j ] m × n

2.2. The element at i The element at i thth row and j row and j thth column is called column is called the (i, j) the (i, j) thth element of the matrix i.e. in element of the matrix i.e. in a a i j i j the first the first subscript i always denotes the number of row subscript i always denotes the number of row and j denotes the number of column in which the and j denotes the number of column in which the element occur.element occur.

3.3. A matrix having 2 rows and 3 columns is of order A matrix having 2 rows and 3 columns is of order 2 2 × 3 and another matrix having 1 row and 2 × 3 and another matrix having 1 row and 2 columns is of order 1 × 2.columns is of order 1 × 2.

Location of the elements in a matrixLocation of the elements in a matrix

For matrix AFor matrix A

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TYPES Of MATRICESTYPES Of MATRICES

MATRICES

ROW MATRIX

COLUMN MATRIX

SQUARE MATRIX

ZERO MATRIX

SYMMTRIC MATRIX

SKEW-SYMMETRIC MATRIX

SQUARE MATRIX

DIAGONAL MATRIX

SCALAR MATRIX

IDNTITY MATRIX

ROW / COLUMN MATRICESROW / COLUMN MATRICES 1. Matrix having only one row is called Row- 1. Matrix having only one row is called Row-

Matrix i.e. the row matrix is of order 1 Matrix i.e. the row matrix is of order 1 × n.× n.

2. Matrix having only one column is called Column-2. Matrix having only one column is called Column-matrix i.e. the column matrix is of order m × 1.matrix i.e. the column matrix is of order m × 1.

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ZERO MATRIXZERO MATRIX

A matrix whose all the elements are A matrix whose all the elements are zero is called zero matrix or null zero is called zero matrix or null matrix and is denoted by O i.e. amatrix and is denoted by O i.e. ai ji j = = 0 for all i, j.0 for all i, j.

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1.1. SQUARESQUARE matrix is a matrix having matrix is a matrix having same number of rows and columns and same number of rows and columns and square matrix having ‘n’ number of square matrix having ‘n’ number of rows and columns is called of order nrows and columns is called of order n

2.2. DIAGONALDIAGONAL matrix is a square matrix if matrix is a square matrix if all its elements except in leading all its elements except in leading diagonal are zero i. e. adiagonal are zero i. e. a ij ij = 0 for i ≠ j = 0 for i ≠ j and aand a ij ij ≠ 0 for i = j. ≠ 0 for i = j.

3.3. SCALARSCALAR matrix is the diagonal matrix matrix is the diagonal matrix with all the elements in leading with all the elements in leading diagonal matrix are same i.e. adiagonal matrix are same i.e. a ij ij = 0 for = 0 for i ≠ j. and ai ≠ j. and a ij ij = k for i = j. = k for i = j.

4.4. UNITUNIT matrix is the scalar matrix with all matrix is the scalar matrix with all the elements in leading diagonal 1 i.e. athe elements in leading diagonal 1 i.e. a

ijij = 0 for i ≠ j. and a = 0 for i ≠ j. and a ij ij = 1 for i = j. = 1 for i = j.

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OPERATION ON MATRICESOPERATION ON MATRICES

Matrices support different basic operationsMatrices support different basic operations..

Some of the basic operations that can be applied areSome of the basic operations that can be applied are

1. Addition of matrices.1. Addition of matrices.

2. Subtraction of matrices.2. Subtraction of matrices.

3. Multiplication of matrices.3. Multiplication of matrices.

4. Multiplication of matrix with scalar value.4. Multiplication of matrix with scalar value.

But two matrices can not be divided.But two matrices can not be divided.

EQUALITY OF MATRICESEQUALITY OF MATRICES

Two matrices are Two matrices are EQUALEQUAL if both are of same if both are of same order and each of the corresponding element order and each of the corresponding element in both the matrices is same. in both the matrices is same.

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ADDITION OF MATRICESADDITION OF MATRICES

Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.

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PROPERTIES OF MATRIX PROPERTIES OF MATRIX ADDITIONADDITION

A + B = B + AA + B = B + A A + ( B + C) = (A + B) + CA + ( B + C) = (A + B) + C A + 0 = 0 + A = AA + 0 = 0 + A = A A + (-A) = 0 = (-A) + AA + (-A) = 0 = (-A) + A A + B = A + C A + B = A + C B = C B = C

MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR

For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.

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PROPERTIES OF SCALAR PROPERTIES OF SCALAR MULTIPLICATIONMULTIPLICATION

k (A + B) = k A + k Bk (A + B) = k A + k B

(-k) A = - (k A) = k (-A)(-k) A = - (k A) = k (-A)

I A = A I = AI A = A I = A

(-1) A = - A(-1) A = - A

MULTIPLICATION OF MATRICESMULTIPLICATION OF MATRICES

Two matrices can be multiplied only if number Two matrices can be multiplied only if number of columns of first is same as number of rows of columns of first is same as number of rows of the second.of the second.

If A is of order m If A is of order m × n and B is of order n × p, × n and B is of order n × p, then the product AB is a matrix of order m ×then the product AB is a matrix of order m × pp..

m m × × n & nn & n × p × p m × pm × p.. For A = [a For A = [a i ji j] ] m×n m×n and B = [ b and B = [ b j kj k] ] n×p n×p , AB = C , AB = C

with C = [cwith C = [cijij] ] m×p m×p where cwhere ci ki k = = ΣΣ a a ij ij b b jkjk

MULTIPLICATION OF MATRICES

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TRANPOSE OF MATRIX

For matrix A = [aij] of order m×n, transpose of A is denoted by AT of A/ and it is a matrix of order n×m and is obtained by interchanging the rows with columns i.e. AT=[aji] with aij = aji for all i,j.

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PROPERTIES OF PROPERTIES OF TRANSPOSE OF MATRICESTRANSPOSE OF MATRICES (A(ATT))TT = A = A (A + B)(A + B)TT= A= ATT + B + BTT

(kA)(kA)TT = k A = k ATT

(AB)(AB)TT = B = BTT A ATT

Every square matrix can be Every square matrix can be expressed as sum of sum of expressed as sum of sum of symmetric and skew-symmetric symmetric and skew-symmetric matrix. A = (A + Amatrix. A = (A + ATT) + (A – A) + (A – ATT))

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SYMMETRIC/SKEW-SYMMETRIC MATRICES

A square matrix A = [aij] is called symmetric matrix if AT = A i.e. aij = aji for all i,j.

A square matrix A = [aij] is called skew-symmetric matrix if AT = -A i.e. aij = - aji for

all i,j.

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IMPORTANT RESULT ON SYMMETRIC AND SKEW-SYMMETRIC MATRICES

Every square matrix can be expressed as sum of sum of symmetric and skew-symmetric matrix. A =(A + AT) +(A – AT)

All the elements in lead diagonal in skew-symmetric matrix are zero.

APPLICATION OF MATRICES

Solution of equations in AX=B system using matrix method

(i) If unique solution with

(ii) If and also (adjA)B = 0, Infinite many solutions.

(iii) If (adj A) B 0 No solution.

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Important Problems

1 Construct a 2 3 matrix A with elements given by

2 Find x, y such that

3 If A = diag.(2 -5 9), B = diag.(1 1 -4), find 3A – 2B.

4 Find X and Y if 2X + Y = and X + 2Y =

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5 Find x if

6 If A = show that

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7 Show that

8 If and Find K so that .

9 Show that is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.

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10 Express A = as sum of symmetric

and skew-symmetric matrices.

11 If A = find

12 Find X if

13 Solve using matrix method

x + 2y + z = 7, x + 3 z = 11, 2 x – 3 y = 1.

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Address of the subject related websites

http://www.netsoc.tcd.ie/~jgilbert/maths_site/applets/algebra/matrix_inversion.html

http://www.ping.be/~ping1339/matr.htm

http://mathworld.wolfram.com/Matrix.html

http://en.wikipedia.org/wiki/Matrices

ACKNOWLEDGEMENT

This power point presentation is prepared under the active guidance of Ms. Summy and Ms. Nidhi the able and learned trainers of project “SHIKSHA” CONDUCTED BY MICROSOFT CORPORATION.

matrices

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