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