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

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#### 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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