Question

Find the matrix A satisfying the matrix equation:

`[(2, 1),(3, 2)] "A" [(-3, 2),(5, -3)] = [(1, 0),(0, 1)]`

Answer 1

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)]`