Question

(
1 3 2
2 1 1
4 0 3
)  = (
7 1 3
1 0 3
−1 −3 7
)

Find the matrix A.

Comments

Please show where A is meant to be in the matrix operation.

Answer 1

A hasn't been shown in the matrix operation, therefore we need to consider all the possibilities for A and all operations.

Let matrix X=

⎛ 1 3 2 ⎞    

⎜ 2 1 1 ⎟

⎝ 4 0 3 ⎠

Let matrix Y=

⎛  7  1 3 ⎞    

⎜  1  0 3 ⎟

⎝ -1 -3 7⎠

ADDITION/SUBTRACTION

X+A=Y, A+X=Y, X=Y-A: A=

⎛  6 -2 1 ⎞    

⎜ -1 -1 2 ⎟

⎝ -5 -3 4 ⎠

X-A=Y, X=A+Y, X=Y+A: A=

⎛ -6 2 -1 ⎞    

⎜  1 1 -2 ⎟

⎝  5 3 -4 ⎠

MULTIPLICATION

First determine X^-1 and Y^-1.

|X|=-11, |Y|=44.

X^-1=

           ⎛   3 -9   1 ⎞    

(-1/11)⎜ -2 -5   3 ⎟

           ⎝ -4 12 -5 ⎠

Y^-1=

           ⎛    9 -16      3 ⎞    

(1/44)⎜ -10   52   -18 ⎟

           ⎝  -3   20     -1 ⎠

If XA=Y then:

X^-1XA=A=X^-1Y=

           ⎛   3 -9   1 ⎞⎛  7  1 3 ⎞  ⎛ -1  0  1 ⎞

(-1/11)⎜ -2 -5   3 ⎟⎜  1  0 3 ⎟=⎜   2  1 0 ⎟

           ⎝ -4 12 -5 ⎠⎝ -1 -3 7⎠   ⎝   1 -1 1 ⎠

This is the likely solution for A (shown in bold type).