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Question
Find the matrix A satisfying the matrix equation:
`[(2, 1),(3, 2)] "A" [(-3, 2),(5, -3)] = [(1, 0),(0, 1)]`
Solution
We have `[(2, 1),(3, 2)] "A" [(-3, 2),(5, -3)] = [(1, 0),(0, 1)]`
or PAQ = I,
Where P = `[(2, 1),(3,2)]` and Q =`[(-3, 2),(5, -3)]`
∴ P–1PAQ = P–1I
⇒ IQA = P–1
⇒ AQ = P–1
⇒ AQQ–1 = P–1Q–1
⇒ AI = P–1Q–1
⇒ A = P–1Q–1
Now adj. P = `[(2, -1),(-3, 2)]` and |P| = 1
∴ P–1 = `[(2, -1),(-3, 2)]`
Also adj . Q = `[(-3, -2),(-5, -3)]` and |Q| = –1
∴ Q–1 = `[(3, 2),(5, 3)]`
⇒ A = P–1Q–1
= `[(2, -1),(-3, 2)][(3, 2),(5, 3)]`
= `[(6 - 5, 4 - 3),(-9 + 10, -6 + 6)]`
= `[(1, 1),(1, 0)]`
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