Question

How do you multiply $3x3$matrix $?$

Answer 1

Let us understand how to perform multiplication of $3x3$ matrices with an example:

Let,

$$\begin{gathered} A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 6& 7\\ 5& 8& 0\end{array}\right] \\ B=\left[\begin{array}{ccc}1& 5& 4\\ 2& 6& 0\\ 3& 8& 7\end{array}\right] \end{gathered}$$

Now,

$AB=$$\left[\begin{array}{ccc}1& 2& 3\\ 4& 6& 7\\ 5& 8& 0\end{array}\mathbf{\right]}$$.\left[\begin{array}{ccc}1& 5& 4\\ 2& 6& 0\\ 3& 8& 7\end{array}\right]$

$\begin{array}{rcl}A{B}_{11}& =& 1\left(1\right)+2\left(2\right)+3\left(3\right)\\ & =& 1+4+9\\ & =& 14\\ A{B}_{12}& =& 1\left(5\right)+2\left(6\right)+3\left(8\right)\\ & =& 5+12+24\\ & =& 41\\ A{B}_{13}& =& 1\left(4\right)+2\left(0\right)+3\left(7\right)\\ & =& 4+0+21\\ & =& 25\\ A{B}_{21}& =& 4\left(1\right)+6\left(2\right)+7\left(3\right)\\ & =& 4+12+21\\ & =& 37\\ A{B}_{22}& =& 4\left(5\right)+6\left(6\right)+7\left(8\right)\\ & =& 20+36+56\\ & =& 112\\ A{B}_{23}& =& 4\left(4\right)+6\left(0\right)+7\left(7\right)\\ & =& 16+0+49\\ & =& 65\\ A{B}_{31}& =& 5\left(1\right)+8\left(2\right)+0\left(3\right)\\ & =& 5+16+0\\ & =& 21\\ A{B}_{32}& =& 5\left(5\right)+8\left(6\right)+0\left(8\right)\\ & =& 25+48+0\\ & =& 73\\ A{B}_{33}& =& 5\left(4\right)+8\left(0\right)+0\left(7\right)\\ & =& 20+0+0\\ & =& 20\end{array}$

Therefore,

$\mathit{A}\mathit{B}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{14}& \mathbf{41}& \mathbf{25}\\ \mathbf{37}& \mathbf{112}& \mathbf{65}\\ \mathbf{21}& \mathbf{73}& \mathbf{20}\end{array}\mathbf{\right]}$

Hence, This method can be used to multiply $3x3$matrix