Is matrix multiplication commutative?

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Is matrix multiplication commutative?

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Answer 1 · 2014-12-11T01:27:15

In general, matrix multiplication is not commutative. There are some exceptions, however, most notably the identity matrices (that is, the n by n matrices $I_{n}$ $I_n$ which consist of 1s along the main diagonal and 0 for all other entries, and which act as the multiplicative identity for matrices)

In general, when taking the product of two matrices $A$ $A$ and $B$ $B$ , where $A$ $A$ is a matrix with $m$ $m$ rows and $n$ $n$ columns and $B$ $B$ is a matrix with $n$ $n$ rows and $p$ $p$ columns, the resultant matrix $A B$ $AB$ will possess $m$ $m$ rows and $p$ $p$ columns. The multiplication cannot occur at all if the number of columns in $A$ $A$ is not equal to the number of rows in $B$ $B$ , so if $A B$ $AB$ exists, the only way for $B A$ $BA$ to exist at all would be if $p = m$ $p=m$ , thus making matrix $A$ $A$ a $m$ $m$ x $n$ $n$ matrix and matrix $B$ $B$ a $n$ $n$ x $m$ $m$ matrix. However, if $p = n$ $p=n$ , it is still quite possible for $A B \neq B A$ $AB != BA$

Recall how each entry in the matrix product is determined. When matrices $A$ $A$ and $B$ $B$ are multiplied into matrix $A B$ $AB$ , each entry in the new matrix is formed from the entries in the old matrix. Specifically, to find matrix entry $A B_{i j}$ $AB_(ij)$ (that is, the entry in the $i$ $i$ row and $j$ $j$ column of matrix $A B$ $AB$ ), we take the dot product of row $i$ $i$ of matrix $A$ $A$ , and column $j$ $j$ of matrix $B$ $B$ .

$A B_{i j} = n \sum k = 1 A_{i k} B_{k j}$ $AB_(ij) = sum_(k=1)^n A_(ik) B_(kj)$

As an example, consider the 3x3 matrices $A$ $A$ and $B$ $B$ , with
$A = ((a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33)))$ and $B = ((b_(11), b_(12), b_(13)),(b_(21), b_(22), b_(23)),(b_(31), b_(32), b_(33)))$ .
Then $AB_(2,3)$ is simply the dot product of the second row of $A$ and the third column of $B$ , or $(a_(21)\ a_(22)\ a_(23))\ .\ ((b_(13)),(b_(23)),(b_(33))) = (a_(21)*b_(13)) + (a_(22)*b_(23)) + (a_(23)*b_(33))$

However, $BA_(2,3)$ would be the dot product of row 2 of matrix $B$ and column 3 of matrix $A$ , or...
$(b_(21)\ b_(22)\ b_(23))\ .\ ((a_(13)),(a_(23)),(a_(33))) = (b_(21)*a_(13)) + (b_(22)* a_(23))+(b_(23)*a_(33))$

If matrix multiplication were commutative, we would expect $BA_(2,3) = AB_(2,3)$ (among other things). Since these expressions may very well not be equal according to our work thus far, we can safely conclude that matrix multiplication is not necessarily commutative.

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Is matrix multiplication commutative?

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