Question

If A = `[(1, -1),(2, 3)], C = [(2, 3),(1, - 1)]`, find matrix B such that BА = С.

Answer 1

Given

A = `[(1, -1),(2, 3)]`

C = `[(2, 3),(1, - 1)]`

BA = C

Step 1: Use the formula

We pre-multiply both sides by A⁻¹

BA = C

B = CA⁻¹

Step 2: Find A⁻¹

For any 2 × 2 matrix A = `[(a, b),(c, d)]` the inverse is

A⁻¹ = `1/(ad - bc) [(d, -b),(-c, a)]`

A = `[(1, -1),(2, 3)]`

So,

a = 1, b = −1, c = 2, d = 3

Determinant = (1) (3) − (−1) (2)

= 3 + 2

= 5

A⁻¹ = `1/5 [(3, 1),(-2, 1)]`

Step 3: Compute B = C. A⁻¹

B = C. A⁻¹

= `[(2, 3),(1, - 1)] xx 1/5 [(3, 1),(-2, 1)]`

= `1/5 [(2 xx 3 + 3 xx (-2), 2 xx 1 + 3 xx 1),(1 xx 3 + (-1) xx (-2), 1 xx 1 + (-1) xx 1)]`

= `1/5 [(6 - 6, 2 + 3),(3 + 2, 1 - 1)]`

= `1/5 [(0, 5),(5, 0)]`

= `[(0, 1),(1, 0)]`