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]]`

Answer 1

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]]`