Question

What is A^(−1)?A=[−2−33 12]Enter your answer by filling in the boxes. Enter any fractions as simplified fractions.A^(−1) =

Answer 1

a11 = -2; a12 = -3; a21 = 3; a22 = 12

We want to find the inverse A^(-1) such as A*A^(-1) = I, where I is the matrix identity.

Let a'11, a12', a21' and a22' be the terms of the matrix A^(-1).

In this case, we have:

I) a11*a11' + a12*a21' = 1

II) a11*a12' + a12*a22' = 0

III) a21*a11' + a22*a21' = 0

IV) a21*a12' + a22*a22' = 1

I) -2a11' - 3a21' = 1

II) -2a12' - 3a22' = 0

III) 3a11' + 12a21' = 0

IV) 3a12' + 12a22' = 1

Adding 4 times equation I to equation III, we have:

-5a11' = 4

a11' = -4/5

Applying this result in equation I, we have:

8/5 -3a21' = 1

-3a21' = -3/5

a21' = 1/5

Adding 4 times equation II to equation IV, we have:

-5a12' = 1

a12' = -1/5

Applying this result in equation II, we have:

2/5 - 3a22' = 1

-3a22' = 3/5

a22' = -1/5

Therefore, the matrix A^(-1) is given by [a11' a12' a21' a22'] = [-4/5 -1/5 1/5 -1/5]