Question

Given A = `[(1, -1),(2, 0)]`, B = `[(3, -2),(1, 1)]` and C = `[(1, 1),(2, 2)]`, find a martix X such that AXB = C

Answer 1

Given A × B × C

⇒ A^–1 A × BB^–1 

= A^–1 C B^–1

I × I = A^–1 CB^–1

⇒ X = A^–1 CB^–1

Let us find A^–1 and B^–1 

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

|A| = 0 + 2

= 2 ≠ 0.A^–1 exists

adjj A = `[(0, 1),(-2, 1)]`

∴ A^–1 = `1/|"A"|` adj A = `1/2[(0, 1),(-2, 1)]`

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

|B| = 3 +2

= 5 ≠ 0.B^–1 exists.

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

∴ B^–1 = `1/|"B"|` adj B = `1/5[(1, 2),(-1, 3)]`

X = A^–1 CB^–1

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

= `1/10 [(0 + 2, 0 + 2),(-2 + 2, -2 + 2)][(1, 2),(-1, 3)]`

= `1/10 [(2, 2),(0, 0)] [(, 2),(-, 3)]`

= `1/10 [(2 - 2, 4 + 6),(0 - 0, 0 - 0)]`

X = `1/10[(0, 10), (0, 0)]`

= `[(0, 1),(0, 0)]`