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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
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &0\ &0\ &\frac{1}{10}\ &-\frac{1}{5}\ \\ &\frac{1}{10}\ &0\ &0\ &\frac{1}{2}\ \\ &0\ &0\ &0\ &\frac{1}{5}\ \\ &0\ &0\ &0\ &0\ \end{pmatrix}\times \begin{pmatrix} &1\ &0\ &0\ &-5\ \\ &0\ &0\ &1\ &-6\ \\ &0\ &-1\ &0\ &3\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}}\\ \end{aligned}
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &0\ &\frac{1}{10}\ &\frac{24}{5}\ \\ &\frac{1}{10}\ &0\ &-1\ &\frac{7}{2}\ \\ &0\ &1\ &0\ &\frac{31}{5}\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &1\ &0\ &0\ &-5\ \\ &0\ &0\ &1\ &-6\ \\ &0\ &-1\ &0\ &3\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}}\\ \end{aligned}
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &1\ &1\ \\ &2\ &2\ &2\ \\ &-1\ &-1\ &-1\ \end{pmatrix}\times \begin{pmatrix} &1\ &1\ &1\ \\ &2\ &2\ &2\ \\ &-1\ &-1\ &-1\ \end{pmatrix}}\\ \end{aligned}
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{433}{500}\ &-\frac{1}{2}\ &0\ \\ &\frac{1}{2}\ &\frac{433}{500}\ &0\ \\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &3\ \\ &5\ \\ &7\ \end{pmatrix}}\\ \end{aligned}
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{3}{2}\ &-\frac{1}{2}\ &0\ \\ &\frac{1}{2}\ &\frac{3}{2}\ &0\ \\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &3\ \\ &5\ \\ &7\ \end{pmatrix}}\\ \end{aligned}
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\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &0\ &\frac{1}{10}\ &\frac{24}{5}\ \\ &\frac{1}{10}\ &0\ &-1\ &\frac{7}{2}\ \\ &0\ &1\ &0\ &\frac{31}{5}\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &1\ &0\ &0\ &-5\ \\ &0\ &0\ &1\ &-6\ \\ &0\ &-1\ &0\ &3\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}}\\ \end{aligned}
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &1\ &1\ \\ &2\ &2\ &2\ \\ &-1\ &-1\ &-1\ \end{pmatrix}\times \begin{pmatrix} &1\ &1\ &1\ \\ &2\ &2\ &2\ \\ &-1\ &-1\ &-1\ \end{pmatrix}}\\ \end{aligned}
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{433}{500}\ &-\frac{1}{2}\ &0\ \\ &\frac{1}{2}\ &\frac{433}{500}\ &0\ \\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &3\ \\ &5\ \\ &7\ \end{pmatrix}}\\ \end{aligned}
\begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{3}{2}\ &-\frac{1}{2}\ &0\ \\ &\frac{1}{2}\ &\frac{3}{2}\ &0\ \\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &3\ \\ &5\ \\ &7\ \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.