Answer to Question #248990 in Linear Algebra for Almas

Question

Prove or Disprove that the commutative property holds on the matrix multiplication.

Answer 1

$\displaystyle \text{Commutative property holds if $MN = NM$, for matrices M and N}\\ \text{Matrix multiplication isn't possible for matrices that aren't square e.g if we have}\\ \text{2 matrices M and N}\\ M_{ij} \cdot N_{jk} \neq N_{jk} \cdot M_{ij}\\ \text{As seen in the right hand side of the expression above the matrix isn't even possible}\\ \text{Hence matrix multiplication is not possible}\\ \text{Also matrix multiplication is not generally possibly for square matrices, to this end}\\ \text{we give a counter-example}\\ \text{Let $ A = \begin{pmatrix}6 & 3\\2 & 5 \end{pmatrix} $ and $B = \begin{pmatrix}-3 & 2\\1 & 5 \end{pmatrix}$}\\ A.B = \begin{pmatrix}-15& 27\\-1 & 29 \end{pmatrix}\\ B.A = \begin{pmatrix}-14 & 1\\16 & 26 \end{pmatrix}\\ \text{Therefore, since $A\cdot B \neq B \cdot A$, hence matrix multiplication isn't commutative}$