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Question
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
`2X + 3Y = [[2,3],[4,0]], 3X+2Y = [[-2,2],[1,-5]]`
Solution
We have,
3(2X+3Y)−2(3X+2Y)=3 `[[2,3],[4,0]]-2 [[-2,2],[1,-5]]`
⇒6X+9Y−6X−4Y= `3[[6,9],[4,0]]+[[4,-4],[-2,10]]`
⇒5Y= `[[6+4,9-4],[12-2,0+10]]`
⇒Y=`1/5 [[10,5],[10,10]]`
⇒Y=`[[2,1],[2,2]]`.............................(1)
Also,
2(2X+3Y)−3(3X+2Y)=2 `[[2,3],[4,0]]-3[[-2,2],[1,-5]]`
⇒4X+6Y−9X−6Y=`[[4,6],[8,0]]+[[6,-6],[-3,15]]`
⇒−5X= `[[6+4,6-6],[8-3,0+15]]`
⇒X=`1/-5[[10,0],[5,15]]`
⇒X= `[[-2,0],[-1,-3]]`...................(2)
From (1) and (2), we get
`X=[[-2,0],[-1,-3]]` And `Y= [[2,1],[2,2]]`
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