Question #248990

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


1
Expert's answer
2021-10-20T01:17:21-0400

Commutative property holds if MN=NM, for matrices M and NMatrix multiplication isn’t possible for matrices that aren’t square e.g if we have2 matrices M and NMijNjkNjkMijAs seen in the right hand side of the expression above the matrix isn’t even possibleHence matrix multiplication is not possibleAlso matrix multiplication is not generally possibly for square matrices, to this endwe give a counter-exampleLet A=(6325) and B=(3215)A.B=(1527129)B.A=(1411626)Therefore, since ABBA, hence matrix multiplication isn’t commutative\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}


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