<< 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!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





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