Question

If A = `[(1, 3),(3, 4)]`, B = `[(-2, 1),(-3, 2)]` and A² – 5B² = 5C. Find matrix C where C is a 2 by 2 matrix.

Answer 1

Given: A = `[(1, 3),(3, 4)]`, B = `[(-2, 1),(-3, 2)]` and A² – 5B² = 5C

Now, A² = A × A 

= `[(1, 3),(3, 4)] xx [(1, 3),(3, 4)]`

= `[(1 xx 1 + 3 xx 3, 1 xx 3 + 3 xx 4),(3 xx 1 + 4 xx 3, 3 xx 3 + 4 xx 4)]`

= `[(1 + 9, 3 + 12),(3 + 12, 9 + 16)]`

= `[(10, 15),(15, 25)]`

And B² = B × B

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

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

= `[(4 - 3, -2 + 2),(6 - 6, -3 + 4)]`

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

Now, A² – 5B² = `[(10, 15),(15, 25)] - 5[(1, 0),(0, 1)]`

= `[(10, 15),(15, 25)] - [(5, 0),(0, 5)]`

= `[(5, 15),(15, 20)]`

= `5[(1, 3),(3, 4)]`

= 5C

Hence, C = `[(1, 3),(3, 4)]`