1. Inverse & Identity Matrices
Section 4.5

2. Objectives You will write the identity matrix for any square matrix
find the inverse of a 2 x 2 matrix

3. The Identity Matrix I
The identity matrix is a square matrix with 1’s on the principal diagonal. All other elements are 0
It’s the only matrix that’s commutative
A • I = I • A = A
Principal Diagonal =
If you multiply the identity matrix is by a 2nd matrix, the product is equal to the second matrix
(it’s like multiplying by 1)
The identity matrix will always be the same dimension as the other matrix.

4. Inverse Matrix A-1
n • 1/n = 1/n • n = 1 is the multiplicative inverse (for real numbers ≠ 0)
A• A-1 = A-1 • A = 1 for matrices if A-1 exists
A 2x2 matrix will have an inverse if its determinant ≠ 0
has the determinant
= ad – cb
If ad = cb, the matrix does not have an inverse

5. Finding A-1 Switch A1,1 & A 2, 2 Find the determinant of the matrix
If A =
Switch A1,1 & A 2, 2
Find the determinant of the matrix 
= ad – cb
If ad – cb = 0, you’re done—no inverse exists
Then change the signs on A1,2 & A2,1 & put the fraction on the left
If ad ≠ cb the matrix has an inverse
Put “1” over the value of the determinant
= A-1

6. find M-1 If M = det of M = Change the matrix to inverse form:
= 14 – 0 = 14
The fraction to use for the inverse is
Change the matrix to inverse form:
(7 & 2 changed places, the sign on the –5 changed to positive & 0 stayed 0)
Set up the inverse:
= M-1

7. Gotta get in some practice!
Find A-1 if A =
(click to check)
(click to check)
Find B-1 if B =
Find C-1 if C =
(click to check)

8. Find A-1 if A =  The determinant: = 4(-1) – 2(3) = -10 
= 4(-1) – 2(3) = -10 
“Everybody Switch! ”
____1____
determinant =
Get the fraction
Put it all together!
_1_
-10 =
= A-1

9. Find B-1 if B = Since B is a 2x3 matrix and is not a square matrix,
It doesn’t have an inverse.

10. Find C-1 if C =
1. Determinant 
= 2 (3) – 1 (6) = 0
C does not have an inverse since the determinant is 0. (Section 4.6 explains why.)