Question

Find the matrix A which satisfies the matrix relation `"A"= [(1, 2, 3),(4, 5, 6)] = [(-7, -8, -9),(2, 4, 6)]`

Answer 1

`"A"= [(1, 2, 3),(4, 5, 6)] = [(-7, -8, -9),(2, 4, 6)]`  ......(1)

(2 × 2)(2 × 3) = 2 × 3

∴ A must be a 2 × 2 matrix.

Let A = `[(x, y),(z, "t")]`

`[(x, y),(z, "t")] [(1, 2, 3),(4, 5, 6)] = [(x + 4y, 2x + 5y, 3x + 6),(z + 4"t", 2z + 5"t", 3z + 6"t")]`

Using eqation (1) we have

`[(x + 4y, 2x + 5y, 3x + 6),(z + 4"t", 2z + 5"t", 3z + 6"t")] = [(-7, -8, -9),(2, 4, 6)]`

Equating the line entries

x + 4y = – 7   .......(1)

2x + 5y = – 8  ......(2)

3x + 6y = – 9   ......(3)

z + 4t = 2   .......(4)  

2z + 5t = 4     ......(5)

3z + 6t = 6   .......(6)

(1) × 2 ⇒ 2x + 8y = – 14
(2)       ⇒ 2x + 5y = –   8
                0  + 3y = –   6

y = `- 6/3`

= – 2

Substituting for y in equation (1)

(1) ⇒ x + 4 × – 2 = – 7
                  × – 8 = – 7
                        x =    8 – 7 = 1
(4) × 2 ⇒ 2z + 8t =   4
(4)       ⇒ 2z + 5t =   4
– ing         0 + 3t =   0

t = `0/3`

= 0

Substituting for t in equation (4)

(4) ⇒ z + 4 × 0 = 2
                      z = 2

∴ The required matrix A is A = `[(1, -2),(2, 0)]`