4.4
Identity and Inverse Matrices
2 x 2 Identity Matrix (I)
1
I = 0
3 x 3 Identity Matrix (I)
1

I = 0
0
0
1
0
1
0
0
0
1
Recall: Typically, AB ≠ BA
Let’s discover why the above matrices are called identity matrices:
3  1
0
− 5
=

 4

2  0
1 

1
0

0
0
1
0
0  − 1
0  2
1   9
4
−3
−2
7 
4  =
0 
If you left or right multiply any identity matrix by any matrix (A) of the
same dimension, you get _________________________.
If two matrices are inverses then:
AB = I
and
BA = I
Determine whether the following matrices are inverses:
 3
A= 
− 5
−1 
2
2
B= 
5
1
3
∴ matrix B is notated as ____________
4.4
Identity and Inverse Matrices
Finding the Inverse of a Matrix
a
Given A = 
c
the inverse of matrix A, A-1 =
3
Example 1: Given A = 
4
1
2
1 d
A − c
b
d 
then
− b
where A ≠ 0
a 
find A-1
Prove that A and the matrix computed for A-1 are truly inverses :
A(A-1) =
AND
(A-1)A =
Example 2:
 6
Given A = 
− 8
 6
Example 3: Given A = 
− 4
1
find A-1 then prove that A and the

− 2
matrix computed for A-1 are truly inverses.
3
− 2
find A-1
4.4
Identity and Inverse Matrices
Solving a Matrix Equation:
AX = B
4 
− 3
3
 5 − 7 X =
2



1.)
Left Multiply
8 
− 2
both sides of the matrix equation by the inverse of A.
2.) What is ( A-1 ) A ?
3.) What must the dimensions of the unknown matrix X be ?
4.) What is the product of the identity matrix and X ?
5.) Find matrix X.
You Try !
Solve the matrix equation AX = B for the 2 x 2 matrix X.
 1
 4

−2 
X =
− 7 
7
− 1

2 
9 
Homework p. 227 #15, 17, 21,22, 26-36 even
4.4
Identity and Inverse Matrices