<< Chapter < Page Chapter >> Page >

Show that the following two matrices are inverses of each other.

A = [ 1 4 −1 −3 ] , B = [ −3 −4 1 1 ]
A B = [ 1 4 −1 −3 ] [ −3 −4 1 1 ] = [ 1 ( −3 ) + 4 ( 1 ) 1 ( −4 ) + 4 ( 1 ) −1 ( −3 ) + −3 ( 1 ) −1 ( −4 ) + −3 ( 1 ) ] = [ 1 0 0 1 ] B A = [ −3 −4 1 1 ] [ 1 4 −1 −3 ] = [ −3 ( 1 ) + −4 ( −1 ) −3 ( 4 ) + −4 ( −3 ) 1 ( 1 ) + 1 ( −1 ) 1 ( 4 ) + 1 ( −3 ) ] = [ 1 0 0 1 ]
Got questions? Get instant answers now!

Finding the multiplicative inverse using matrix multiplication

We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication .

Finding the multiplicative inverse using matrix multiplication

Use matrix multiplication to find the inverse of the given matrix.

A = [ 1 −2 2 −3 ]

For this method, we multiply A by a matrix containing unknown constants and set it equal to the identity.

[ 1 −2 2 −3 ]     [ a b c d ] = [ 1 0 0 1 ]

Find the product of the two matrices on the left side of the equal sign.

[ 1 −2 2 −3 ]     [ a b c d ] = [ 1 a −2 c 1 b −2 d 2 a −3 c 2 b −3 d ]

Next, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the identity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding entry of the identity, which is 0.

1 a −2 c = 1      R 1 2 a −3 c = 0      R 2

Using row operations, multiply and add as follows: ( −2 ) R 1 + R 2 R 2 . Add the equations, and solve for c .

1 a 2 c = 1 0 + 1 c = 2 c = 2

Back-substitute to solve for a .

a −2 ( −2 ) = 1 a + 4 = 1 a = −3

Write another system of equations setting the entry in row 1, column 2 of the new matrix equal to the corresponding entry of the identity, 0. Set the entry in row 2, column 2 equal to the corresponding entry of the identity.

1 b −2 d = 0 R 1 2 b −3 d = 1 R 2

Using row operations, multiply and add as follows: ( −2 ) R 1 + R 2 = R 2 . Add the two equations and solve for d .

1 b −2 d = 0 0 + 1 d = 1 d = 1

Once more, back-substitute and solve for b .

b −2 ( 1 ) = 0 b −2 = 0 b = 2
A −1 = [ −3 2 −2 1 ]
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the multiplicative inverse by augmenting with the identity

Another way to find the multiplicative inverse is by augmenting with the identity. When matrix A is transformed into I , the augmented matrix I transforms into A −1 .

For example, given

A = [ 2 1 5 3 ]

augment A with the identity

[ 2 1 5 3   |   1 0 0 1 ]

Perform row operations    with the goal of turning A into the identity.

  1. Switch row 1 and row 2.
    [ 5 3 2 1   |   0 1 1 0 ]
  2. Multiply row 2 by −2 and add to row 1.
    [ 1 1 2 1   |   −2 1 1 0 ]
  3. Multiply row 1 by −2 and add to row 2.
    [ 1 1 0 −1   |   −2 1 5 −2 ]
  4. Add row 2 to row 1.
    [ 1 0 0 −1   |   3 −1 5 −2 ]
  5. Multiply row 2 by −1.
    [ 1 0 0 1   |   3 −1 −5 2 ]

The matrix we have found is A −1 .

A −1 = [ 3 −1 −5 2 ]

Finding the multiplicative inverse of 2×2 matrices using a formula

When we need to find the multiplicative inverse of a 2 × 2 matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.

If A is a 2 × 2 matrix, such as

A = [ a b c d ]

the multiplicative inverse of A is given by the formula

A −1 = 1 a d b c [ d b c a ]

where a d b c 0. If a d b c = 0 , then A has no inverse.

Using the formula to find the multiplicative inverse of matrix A

Use the formula to find the multiplicative inverse of

A = [ 1 −2 2 −3 ]

Using the formula, we have

A −1 = 1 ( 1 ) ( −3 ) ( −2 ) ( 2 ) [ −3 2 −2 1 ] = 1 −3 + 4 [ −3 2 −2 1 ] = [ −3 2 −2 1 ]
Got questions? Get instant answers now!
Got questions? Get instant answers now!
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask