### Inverse

##### 3. Inverse of 2 by 2 matrix

A 1 by 1 matrix A = (a) is invertible if and only if a ≠ 0. Moreover, the inverse A-1 = (a-1).

Assume the 2 by 2 matrix

 A = [ a b ] c d

has

 B = [ x z ] y w

as the inverse. Then from AB = I we have

ax + by = 1, cx + dy = 0, az + bw = 0, cz + dw = 1.

The first two is a system of linear equations with A as the coefficient matrix and x, y as variables. Then d[equation 1] - b[equation 2] and - c[equation 1] + a[equation 2] give us

Similarly, from the 3rd and the 4th equations we get

Thus we conclude the ⇐ direction of the following result.

 A 2 by 2 matrix [ a b ] is invertible ⇔ ad - bc ≠ 0. Moreover, the inverse is c d
 1 [ d -b ] ad - bc -c a

For the ⇒ direction, see this exercise.

Example We have

 [ 1 3 ]-1 = (1×4 - 3×2)-1[ 4 -3 ]-1 = [ -2 3/2 ] 2 4 -2 1 1 -1/2

We verify our computation as follows.

 [ 1 3 ] [ -2 3/2 ] = [ 1×(-2) + 3×1 1×3/2 + 3×(-1/2) ] = [ 1 0 ] 2 4 1 -1/2 2×(-2) + 4×1 2×3/2 + 4×(-1/2) 0 1
 [ -2 3/2 ] [ 1 3 ] = [ (-2)×1 + (3/2)×2 (-2)×3 + (3/2)×4 ] = [ 1 0 ] 1 -1/2 2 4 1×1 + (-1/2)×2 1×3 + (-1/2)×4 0 1

The inverse also tells us that the system

 x1 + 3x2 = 1 2x1 + 4x2 = -4

has

 [ x1 ] = [ 1 3 ]-1 [ 1 ] = [ -2 3/2 ] [ 1 ] = [ 4 ] x2 2 4 -4 1 -1/2 4 -1

as the unique solution.

Example The following matrices

 [ 2 4 ] -1 -2
 [ 1 5 ] 3 15

are not invertible because 2×(-2) - 4×(-1) = 0, 1×15 - 5×3 = 0.

The formula for the inverse of 2 by 2 matrices can be generalized to bigger matrices. However, the amount of computation involved in such a formula will become too big to be practical.