MHB Find Matrix A: Determining Inverse of Matrices

[lyd123]

Hi, I've a question that asks me to determine matrix A , where A= ${S}^{-1}$* B* S
They have given matrices S and B in the question.

I think the answer is A=B, since S * ${S}^{-1}$ would give me the identity matrix and anything multiplied by the identity matrix is itself. Is this correct?

 

[Olinguito]

Hi lyd123.

The answer $A=B$ would be correct only if the matrices $B$ and $S$ commute, i.e. only if $BS=SB$. Perhaps the matrices you are given commute. In general, however, matrix multiplication is not commutative. Given
$$A\ =\ S^{-1}BS$$
the best you can do is multiply on the left by $S$ and on the right by $S^{-1}$ to get
$$SAS^{-1}\ =\ S(S^{-1}BS)S^{-1}\ =\ (SS^{-1})B(SS^{-1})\ =\ IBI\ =\ B.$$

 

[lyd123]

Thank you! I totally forgot that matrix multiplication isn't commutative.