# Given A = 4 -1 , B = 6 4 , C= 1 2 -2 3 -5 -3 3 4 and APB = C, find the matrix P.Please A, B and C are matrices...

Given A = 4 -1 , B = 6 4 , C= 1 2

-2 3 -5 -3 3 4

and APB = C, find the matrix P.

Please A, B and C are matrices but I was not able to put brackets around them.

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### 1 Answer

Given matrices whose entries are: `A:a_(1,1)=4,a_(1,2)=-1,a_(2,1)=-2,a_(2,2)=3`

`B:b_(1,1)=6,b_(1,2)=4,b_(2,1)=-5,b_(2,2)=-3`

`C:c_(1,1)=1,c_(1,2)=2,c_(2,1)=3,c_(2,2)=4`

Find matrix `P` such that `APB=C` .

(1) Clearly `P` is a 2x2 matrix, so let the elements be `p_(1,1)=a, p_(1,2)=b,p_(2,1)=c,p_(2,2)=d`

(2) Multiplying left to right, we take `AP` first. (Matrix multiplication is associative, so you could take `PB` first if you want. Your choices are `(AP)B` or `A(PB)` )

So `AP` is a 2x2 matrix with entries (1,1)=4a-c,(1,2)=4b-d,(2,1)=-2a+3c,(2,2)=-2b+3d

(3) Now `(AP)B` gives :

(1,1)=24a-6c-20b+5d

(1,2)=16a-4c-12b+3d

(2,1)=-12a+18c+10b-15d

(2,2)=-8a+12c+6b-9d

(4) Equating the corresponding entries with those of `C` we get the folowing system of equations:

`24a-20b-6c+5d=1`

`16a-12b-4c+3d=2`

`-12a+10b+18c-15d=3`

`-8a+6b+12c-9d=4`

(5) There are a number of ways to solve this system. Using Gaussian elimination to put the augmented matrix in reduced row-echelon form you get:

`a=8/5=1.6`

`b=9/5=1.8`

`c=2.9=29/10`

`d=16/5=3.2`

You could also use linear combinations, substitution, or matrix multiplication if you can find the inverse of the 4x4 matrix.

(6)** Thus the matrix `P` has entries:**

`p_(1,1)=1.6,p_(1,2)=1.8,p_(2,1)=2.9,p_(2,2)=3.2`