# Solve for X. [[-4,-2],[2,0]] dot X + [[1,1],[0,-4]] = X dot [[0,1],[1,2]] X = ??

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### 2 Answers

Here because our matrix `[[-4,-2],[2,0]]` is 2by2. So our vector X should be `X=[x_1,x_2]^T` .

Now we see that the `X.[[0,1],[1,2]]` are not conformable for multiplication.

So in order to find the value of X the multiplication should be same sided i.e. of AX=BX form only then we can find the value of the vector X by comparing similar elements.

[[-4,-2],[2,0]] dot X + [[1,1],[0,-4]] = X dot [[0,1],[1,2]]

`[[-4,-2],[2,0]].X+[[1,1],[0,-4]]=X.[[0,1],[1,2]]` (i)

`[[0,1],[1,2]]^(-1)=(-1)[[2,-1],[-1,0]]`

=`[[-2,1],[1,0]]` (ii)

Post multiply (i) by (ii) ,we have

`[[-4,-2],[2,0]]X[[-2,1],[1,0]]+[[1,1],[0,-4]][[-2,1],[1,0]]=X` (iii)

Let `X=[[a,b],[c,d]]`

substitute X in (iii) and simplify we have

`[[-8a-4b-4c-2d-1,-4a-2b+1],[-4a+2c-4,2a]]=[[a,b],[c,d]]`

`2a=d`

`-8a-4b-4c-2d-1=a`

`-4a-2b+1=b`

`-4a+2c-4=c`

solving above system of equations in a,b,c and d. we have

`a=-55/71,b=97/71,c=64/71,d=91/71`

`X=[[-55/71,97/71],[64/71,91/71]]`

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