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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
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &92\ &61\ \\ &190\ &186\ \end{pmatrix}\times \begin{pmatrix} &3\ \\ &4\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ \\ &2\ \\ &3\ \\ &4\ \\ &5\ \\ &6\ \\ &7\ \\ &8\ \\ &9\ \end{pmatrix}\times \begin{pmatrix} &1\ &2\ &3\ &4\ &5\ &6\ &7\ &8\ &9\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &7\ &0\ &0\ \\ &2\ &3\ &1\ \\ &6\ &1\ &1\ \end{pmatrix}\times \begin{pmatrix} &1\ &2\ &2\ \\ &1\ &0\ &1\ \\ &-5\ &4\ &1\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &-\frac{3}{100}\ &0\ &-\frac{1}{10}\ &-\frac{79}{100}\ \\ &0\ &\frac{3}{100}\ &0\ &\frac{9}{100}\ \\ &0\ &-\frac{1}{10}\ &0\ &-\frac{2}{5}\ \\ &0\ &0\ &0\ &0\ \end{pmatrix}\times \begin{pmatrix} &0\ &1\ &0\ &-2\ \\ &1\ &0\ &0\ &-3\ \\ &0\ &0\ &-1\ &8\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &0\ &1\ &0\ &2\ \\ &1\ &0\ &0\ &3\ \\ &0\ &0\ &-1\ &-8\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &-\frac{3}{100}\ &0\ &-\frac{1}{10}\ &-\frac{79}{100}\ \\ &0\ &\frac{3}{100}\ &0\ &\frac{9}{100}\ \\ &0\ &-\frac{1}{10}\ &0\ &-\frac{2}{5}\ \\ &0\ &0\ &0\ &0\ \end{pmatrix}}\\ \end{aligned}$$
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$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ \\ &2\ \\ &3\ \\ &4\ \\ &5\ \\ &6\ \\ &7\ \\ &8\ \\ &9\ \end{pmatrix}\times \begin{pmatrix} &1\ &2\ &3\ &4\ &5\ &6\ &7\ &8\ &9\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &7\ &0\ &0\ \\ &2\ &3\ &1\ \\ &6\ &1\ &1\ \end{pmatrix}\times \begin{pmatrix} &1\ &2\ &2\ \\ &1\ &0\ &1\ \\ &-5\ &4\ &1\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &-\frac{3}{100}\ &0\ &-\frac{1}{10}\ &-\frac{79}{100}\ \\ &0\ &\frac{3}{100}\ &0\ &\frac{9}{100}\ \\ &0\ &-\frac{1}{10}\ &0\ &-\frac{2}{5}\ \\ &0\ &0\ &0\ &0\ \end{pmatrix}\times \begin{pmatrix} &0\ &1\ &0\ &-2\ \\ &1\ &0\ &0\ &-3\ \\ &0\ &0\ &-1\ &8\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &0\ &1\ &0\ &2\ \\ &1\ &0\ &0\ &3\ \\ &0\ &0\ &-1\ &-8\ \\ &0\ &0\ &0\ &1\ \end{pmatrix}\times \begin{pmatrix} &-\frac{3}{100}\ &0\ &-\frac{1}{10}\ &-\frac{79}{100}\ \\ &0\ &\frac{3}{100}\ &0\ &\frac{9}{100}\ \\ &0\ &-\frac{1}{10}\ &0\ &-\frac{2}{5}\ \\ &0\ &0\ &0\ &0\ \end{pmatrix}}\\ \end{aligned}$$
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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.