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Question
Find the matrix X so that X`[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]`
Solution
X = `[(a,b), (c,d)]`
`"X"[(1,2,3), (4,5,6)] = [(a,b), (c,d)][(1,2,3), (4,5,6)]`
= `[(a + 4b,2a + 5b,3a + 6b), (c + 4d, 2c + 5d, 3c + 6d)]`
= `[(-7,-8,-9), (2,4,6)]` ...(Given)
Keeping corresponding elements same,
a + 4b = -7 ...(1)
2a + 5b = -8 ...(2)
- - +
3a + 6b = -9 ...(3)
Multiplying equation (1) by 2 and subtracting it from equation (2),
2a + 8b = -14
2a + 5b = -8
- - +
3b = -6
b = -2
Putting the value of b in equation (3),
3a + 6 × (-2) = -9
3a - 12 = -9
3a = 12 - 9 = 3
a = 1
Keeping the corresponding elements of the second row same,
c + 4d = 2 ...(4)
2c + 5d = 4 ...(5)
3c + 6d = 6 ...(6)
On multiplying equation (4) by 2 and subtracting it from equation (5), we get
2c + 8d = 4
2c + 5d = 4
- - +
3d = 0
d = 0
Putting the value of d in equation (6),
3c = 6
c = 2
Hence, X = `[(1, -2), (2, 0)]`
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