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Matrix multiplication:
Enter two matrices that can be multiplied, with each element separated by a comma and each row ending with a semicolon.
Note that mathematical functions and variables are not supported.
Enter two matrices that can be multiplied, with each element separated by a comma and each row ending with a semicolon.
Note that mathematical functions and variables are not supported.
Current location：Linear algebra >Matrix multiplication >History of matrix multiplication >Answer
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &-1\ &3\ \\ &4\ &3\ &2\ \end{pmatrix}\times \begin{pmatrix} &-\frac{1}{3}\ &\frac{2}{3}\ &-\frac{1}{3}\ \\ &\frac{2}{3}\ &-\frac{1}{3}\ &\frac{2}{3}\ \\ &-1\ &1\ &0\ \end{pmatrix}}\\ \\Solution:&\\&\begin{pmatrix} &1\ &-1\ &3\ \\ &4\ &3\ &2\ \end{pmatrix}\times \begin{pmatrix} &-\frac{1}{3}\ &\frac{2}{3}\ &-\frac{1}{3}\ \\ &\frac{2}{3}\ &-\frac{1}{3}\ &\frac{2}{3}\ \\ &-1\ &1\ &0\ \end{pmatrix}\\\\=\ \ &\begin{pmatrix} &-4\ &4\ &-1\ \\ &-\frac{4}{3}\ &\frac{11}{3}\ &\frac{2}{3}\ \end{pmatrix}\end{aligned}$$
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The properties of matrix multiplication:
(i) Combining Law: （A b）C=A（b C）
(ii) Distribution Law: A ( B + C ) = A B + A C either or ( A + B ) C = A C + B C .
(iii) λ ( A B ) = ( λ A ) B = A ( λ B ) .
Among them, A, B, and C are the matrices that make the multiplication of the above matrices meaningful, λ It's a number.