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