Inverse & IdentityMatrices

Section 4.5

Objectives

You will

• write the identity matrix for any square matrix

• find the inverse of a 2 x 2 matrix

The Identity Matrix IThe 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

1 2 4

0 9 8

7 5 3

1 0 0

0 1 0

0 0 1

=1 2 4

0 9 8

7 5 3

The identity matrix will always be the same dimension as the other matrix.

1 0

0 1

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)

Principal Diagonal

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

a b

c d

has the determinanta b

c d= ad – cb

If ad = cb, the matrix does not have an inverse

Finding A-1

Find the determinant of the matrix

Switch A1,1 & A 2, 2

a b

c d= ad – cb

If ad – cb = 0, you’re done—no inverse exists

a b

c d

If ad ≠ cb the matrix has an inverse

1

ad cb

Put “1” over the value of the determinant

If A =

a b

c d

d b

c a

Then change the signs on A1,2 & A2,1 & put the fraction on the left

d b

c a

1

ad cb = A-1

find M-1

det of M =

2 5

0 7

If M =

2 5

0 7

= 14 – 0 = 14

The fraction to use for the inverse is1

14

Change the matrix to inverse form: 7 5

0 2

(7 & 2 changed places, the sign on the –5

changed to positive & 0 stayed 0)

Set up the inverse: 1

14

7 5

0 2

= M-1

Gotta get in some practice!

• Find A-1 if A = 4 3

2 1

(click to check)• Find B-1 if B =

1 2 4

3 0 5

(click to check)

• Find C-1 if C = (click to check)2 6

1 3