Question

Find the matrix A for which A`[(5, 3),(-1, -2)] = [(14, 7),(7, 7)]`

Answer 1

Given A`[(5, 3),(-1, -2)] = [(14, 7),(7, 7)]`

Let B = `[(5, 3),(-1, -2)]`

C = `[(14, 7),(7, 7)]`

Given AB = C,

To find A:

Now AB = C

Post multiply by B^–1 on both sides

ABB^–1 = CB^–1 

(i.e) A(BB^–1) = CB^–1

⇒ A(I) = CB^–1 

(i.e) A = CB^–1

To find B^–1:

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

|B| = `|(5, 3),(-1, -2)|`

= – 10 + 3

= – 7 ≠ 0

adj B = `[(-2, -3),(1, 5)]`

B^–1 = `1/|"B"|`

(adj B) = `1/(-7)[(-, -3),(1, 5)]`

= `1/7 [(2, 3),(-1, -5)]`

A = CB^–1 = `1/7 [(14, 7),(7, 7)] [(2, 3),(-1, -5)]` 

= `1/7 (7) [(2, 1),(1, 1)] (2, 3),(-1, -5)]`

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

= `[(4 - 1, 6 - 5), (2 - 1, 3 - 5)]`

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

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