finding the inverse of a matrix. properties of matrices we have discovered that the commutative...
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Finding the Inverse of a Matrix
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Properties of Matrices
We have discovered that the commutative We have discovered that the commutative property for multiplication does not work for property for multiplication does not work for matrix multiplication. Let’s consider some matrix multiplication. Let’s consider some of the other properties of real numbers. Is of the other properties of real numbers. Is there a multiplicative identity for matrices? there a multiplicative identity for matrices? Is there a multiplicative inverse for Is there a multiplicative inverse for matrices?matrices?
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The Multiplicative Identity
The multiplicative identity for real numbers The multiplicative identity for real numbers is the number 1. The property is:is the number 1. The property is:
In terms of matrices we need a matrix In terms of matrices we need a matrix that can be multiplied by a matrix (A) and that can be multiplied by a matrix (A) and give a product which is the same matrix give a product which is the same matrix (A).(A).
If a is a real number, then a x 1 = 1 x a = a.If a is a real number, then a x 1 = 1 x a = a.
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The Multiplicative Identity
10
012I
This matrix exists and it is called the This matrix exists and it is called the identity matrix. It is named identity matrix. It is named II and it comes and it comes in different sizes. It is a square matrix with in different sizes. It is a square matrix with all 1’s on the main diagonal and all other all 1’s on the main diagonal and all other entries are 0.entries are 0.
100
010
001
3I
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The Multiplicative Identity
04
52A
10
01
04
52Multiply Multiply AAII
aa1111= (-2)(1) + (5)(0) = -2 = (-2)(1) + (5)(0) = -2
aa1212= (-2)(0) + (5)(1) = 5 = (-2)(0) + (5)(1) = 5
aa2121= (4)(1) + (0)(0) = 4 = (4)(1) + (0)(0) = 4
aa2222= (4)(0) + (0)(1) = 0= (4)(0) + (0)(1) = 0
04
52
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The Identity Matrix for Multiplication
Let Let AA be a square matrix with n rows be a square matrix with n rows and n columns. Let and n columns. Let II be a matrix with be a matrix with the same dimensions and with 1’s on the same dimensions and with 1’s on the main diagonal and 0’s elsewhere. the main diagonal and 0’s elsewhere.
Then Then AAII = = IIA = AA = A
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The Multiplicative Identity
1406
7410
2973
9470
B
Give the multiplicative identity for matrix B.Give the multiplicative identity for matrix B.
1000
0100
0010
0001
I
This identity matrix is This identity matrix is II4.4.
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The Multiplicative Inverse
For every nonzero real number a, there is a For every nonzero real number a, there is a real number 1/a such that a(1/a) = 1.real number 1/a such that a(1/a) = 1.
In terms of matrices, the product of a In terms of matrices, the product of a square matrix and its inverse is square matrix and its inverse is II..
10
01
)3(1)1(2)2(1)1(2
)3(1)1(3)2(1)1(3
32
11
12
13
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The Inverse of a Matrix
Let Let AA be a square matrix with be a square matrix with nn rows rows and and nn columns. If there is an columns. If there is an nn x x nn matrix matrix BB such that such that AB = AB = II and and BA = BA = II, , then then AA and and BB are inverses of one are inverses of one another. The inverse of matrix another. The inverse of matrix AA is is denoted by denoted by AA-1-1..
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The Inverse of a Matrix
To show that matrices are inverses of one To show that matrices are inverses of one another, show that the multiplication of the another, show that the multiplication of the matrices is commutative and results in the matrices is commutative and results in the identity matrix.identity matrix.
Show that A and B are inverses.Show that A and B are inverses.
23
35
53
32BandA
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The Inverse of a Matrix
10
01
)2(5)3(3)3(5)5(3
)2(3)3(2)3(3)5(2
23
35
53
32AB
and and
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The Inverse of a Matrix
10
01
)5(2)3(3)3(2)2(3
)5)(3()3(5)3)(3()2(5
53
32
23
35BA
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Finding the Inverse of a Matrix - Method 1
dc
baBandALet
53
21Use the equation AB = Use the equation AB = I.I.
Write and solve the equation:Write and solve the equation:
10
01
53
21
dc
ba
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Inverses – Method 1, cont.
10
01
53
21
dc
ba
10
01
5353
22
dbca
dbca
1235
153
02
053
12
dandbcanda
db
db
ca
ca
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Inverses – Method 1, cont.
13
25So the inverse of A = So the inverse of A =
We can check this by multiplying A x AWe can check this by multiplying A x A-1-1
10
01
)1(5)2(3)3(5)5(3
)1(2)2(1)3(2)5(1
13
25
53
21
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Finding the Inverse with a Calculator
To find the inverse of a matrix using the To find the inverse of a matrix using the calculator, enter the matrix into the calculator, enter the matrix into the calculator and use the xcalculator and use the x-1-1 key. key.
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Finding the Inverse with a Calculator
012
431
112
B
36
48C
Find the inverse of each matrix using the Find the inverse of each matrix using the calculator.calculator.
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Finding the Inverse with a Calculator
This error message This error message means that the matrix means that the matrix does not have an does not have an inverse.inverse.
A matrix that does not have an inverse is A matrix that does not have an inverse is called an called an invertibleinvertible matrix. matrix.
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DeterminantsDeterminants
Each square matrix can be Each square matrix can be assigned a real number called assigned a real number called the the determinantdeterminant of the matrix. of the matrix. It is denoted by the symbol It is denoted by the symbol ..
dc
ba
dc
baAIf
means the means the determinant determinant of A.of A.
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DeterminantsDeterminants
The determinant of a 2 x 2 The determinant of a 2 x 2 matrix is found as follows:matrix is found as follows:
cbaddc
ba
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Determinants
76
87GFind the determinant Find the determinant
of the matrix.of the matrix.
14849)8(6)7(776
87
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Determinants
22
11H
Find the determinant of the matrix.Find the determinant of the matrix.
0)1(2)2(122
11
If the determinant If the determinant of a matrix = 0, the of a matrix = 0, the matrix does not matrix does not have an inverse. have an inverse. Matrix H is Matrix H is invertible.invertible.
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Determinants can be used to find the inverse of a matrix.
ac
bd
AdetA
thenAdetanddc
baAIf
)(
1
,0)(
1
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Determinants can be used to find the inverse of a matrix.
ac
bdis called the adjoint of the original matrix. Notice it is matrix. Notice it is
found by switching the entries on the main diagonal and changing the signs of the entries on the other diagonal.
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Find the multiplicative Find the multiplicative inverse of:inverse of:
43
21A
21
23
12
13
24
2
11A
2)2(3)4(143
21
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We can check to see if we are correct We can check to see if we are correct by multiplying. Remember that by multiplying. Remember that AAAA-1-1 = = II
10
01
)2/1(4)1(3)2/3(4)2(3
)2/1(2)1(1)2/3(2)2(1
21
23
12
43
21
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Find the inverse using determinants.
11
31
21
21
23
21
30
12
31061
21
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Find the inverseFind the inverse
No No inverseinverse
Recall that when the Recall that when the determinant of a matrix is 0 determinant of a matrix is 0 the matrix will not have an the matrix will not have an inverse because division by 0 inverse because division by 0 is undefined.is undefined.
42
84
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Finding the determinant of a 3 x 3 matrix
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Finding the determinant of a 3x3 matrix.
hg
edc
ig
fdb
ih
fea
ihg
fed
cba
One way to find the determinant of a 3x3 One way to find the determinant of a 3x3 matrix is the formula below.matrix is the formula below.
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Find the determinant using the formula
420
513
502
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Find the determinant using the formula
2
3028
)6(5)12(0)14(2
)1(0)2(35)5(0)4(30)5(2)4(12
20
135
40
530
42
512
420
513
502
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Find the determinant using the formula
214
321
112
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Find the determinant using the formula
13
91014
)9(1)10(1)7(2
)2(4)1(11)3(4)2(11)3(1)2(22
14
211
24
31)1(
21
322
214
321
112