Multiplication of Matrices:
We can only multiply matrices if the number of columns in the first matrix is the same as the number of rows in the second matrix.
Multiplication of 2x2 matrix:
The process is the same for any size matrix. We multiply across rows of the first matrix and down columns of the second matrix, element by element. We then add the products:
Given A=\[\begin{bmatrix} 8&9 \\ 5 &-1 \\ \end{bmatrix}\] and B=\[\begin{bmatrix} -2&3 \\ 4 &0 \\ \end{bmatrix}\] lets find AxB.
=\[\begin{bmatrix} 8 & 9 \\ 5 & -1 \\ \end{bmatrix} \times \begin{bmatrix} -2&3 \\ 4 &0 \\ \end{bmatrix}\]
=\[\begin{bmatrix} 8\times−2+9\times4 & 8\times3+9\times0 \\ 5\times-2+ -1\times4&5\times3+-1\times0 \\ \end{bmatrix}\]
=\[\begin{bmatrix} -16+36 & 24+0\\ -10+-4 & 15+0 \\ \end{bmatrix}\]
=\[\begin{bmatrix} 20& 24\\ -16 & 15 \\ \end{bmatrix}\]
Multiplication of 3x3 matrix:
The process is the same for any size matrix. We multiply across rows of the first matrix and down columns of the second matrix, element by element. We then add the products:
Given A=\[\begin{bmatrix} 4&6 & 8\\ 2 & 1 & 4 \\ 6& 9 & 3\\ \end{bmatrix}\] and B=\[\begin{bmatrix} 1&5 & 8\\ 4 & 3& 2 \\ 7& 6 & 5\\ \end{bmatrix}\] lets find AxB.
=\[\begin{bmatrix} 4&6 & 8\\ 2 & 1 & 4 \\ 6& 9 & 3\\ \end{bmatrix}\]x\[\begin{bmatrix} 1&5 & 8\\ 4 & 3& 2 \\ 7& 6 & 5\\ \end{bmatrix}\]
=\[\begin{bmatrix} 4\times1+6\times4+8\times7 & 4\times5+6\times3+8\times6 &4\times8+6\times2+8\times5 \\ 2\times1+1\times4+4\times7& 2\times5+1\times3+4\times6 & 2\times8+1\times2+4\times5 \\ 6\times1+9\times4+3\times7& 6\times5+9\times3+3\times6 & 6\times8+9\times2+3\times5\\ \end{bmatrix}\]
=\[\begin{bmatrix} 84& 86 &84 \\ 34 &37 &38 \\ 63 & 75& 81\\ \end{bmatrix}\]
Multiplication of 4x4 matrix:
The process is the same for any size matrix. We multiply across rows of the first matrix and down columns of the second matrix, element by element. We then add the products:
Given A=\[\begin{bmatrix} 3& 5 & 7 & 8 \\ 1& 2& 8 & 7\\ 4& 5 &3 & -2 \\ 1& 6 & 7 &9 \\ \end{bmatrix}\] and B= \[\begin{bmatrix} 2& 8 & 6 & 9 \\ 3& -5& 6 & 7\\ 1& 4 &9 & -3 \\ 10& -2 & 5 &92\\ \end{bmatrix}\]
=\[\begin{bmatrix} 108& 11& 151 &57 \\ 86& 16 &125 & 13\\ 6& 23 & 71 & 58 \\ 117 & -12 & 150 & 48 \\ \end{bmatrix}\]
