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Question
Let A = `((2,-1),(3,4))`, B = `((5,2),(7,4))`, C= `((2,5),(3,8))` find a matrix D such that CD − AB = O
Solution
Since A, B and C are all square matrices or order 2 and CD − AB is well defined, D must be a square matrix of order 2.
Let D = `[(a,b),(c,d)]`
Then CD − AB = O gives,
`[(2,5),(3,8)][(a,b),(c,d)] - [(2,-1),(3,4)][(5,2),(7,4)] = O`
`=>[(2a + 5c,2b+5a),(3a+8c, 3b+8a)]-[(3,0),(43,22)] = [(0,0),(0,0)]`
`=>[(2a + 5c - 3, 2b + 5d),(3a + 8c - 43, 3b + 8d - 22)] = [(0,0),(0,0)]`
By equality of matrices we get,
2a + 5c − 3 = 0 ...(1)
3a + 8c − 43 = 0 ...(2)
2b + 5d = 0 ...(3)
3b + 8d − 22 = 0 ...(4)
By solving (1) and (2) we get a = −191 and c = 77.
Similarly, on solving (3) and (4) we get b = - 110 and d = 44.
Therefore,
`D = [(a,b),(c,d)] = [(-191,-110),(77,44)]`
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