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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
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &8\ &6\ &9\ &3\ &6\ &8\ &4\ \\ &7\ &3\ &6\ &2\ &1\ &7\ &5\ &8\ \\ &2\ &8\ &6\ &9\ &4\ &5\ &2\ &1\ \\ &4\ &2\ &9\ &7\ &7\ &6\ &8\ &2\ \\ &1\ &8\ &3\ &8\ &5\ &9\ &4\ &2\ \\ &9\ &5\ &9\ &4\ &3\ &7\ &4\ &2\ \\ &3\ &9\ &6\ &4\ &3\ &9\ &6\ &5\ \\ &1\ &5\ &3\ &8\ &6\ &4\ &2\ &8\ \end{pmatrix}\times \begin{pmatrix} &1\ &8\ &6\ &9\ &3\ &6\ &8\ &4\ \\ &7\ &3\ &6\ &2\ &1\ &7\ &5\ &8\ \\ &2\ &8\ &6\ &9\ &4\ &5\ &2\ &1\ \\ &4\ &2\ &9\ &7\ &7\ &6\ &8\ &2\ \\ &1\ &8\ &3\ &8\ &5\ &9\ &4\ &2\ \\ &9\ &5\ &9\ &4\ &3\ &7\ &4\ &2\ \\ &3\ &9\ &6\ &4\ &3\ &9\ &6\ &5\ \\ &1\ &5\ &3\ &8\ &6\ &4\ &2\ &8\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \end{pmatrix}\times \begin{pmatrix} &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \end{pmatrix}\times \begin{pmatrix} &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &0\ \\ &21\ \end{pmatrix}\times \begin{pmatrix} &0\ &21\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \end{pmatrix}\times \begin{pmatrix} &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \end{pmatrix}}\\ \end{aligned}$$
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$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \end{pmatrix}\times \begin{pmatrix} &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ &\frac{1}{5}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \end{pmatrix}\times \begin{pmatrix} &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ &\frac{1}{4}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &0\ \\ &21\ \end{pmatrix}\times \begin{pmatrix} &0\ &21\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \end{pmatrix}\times \begin{pmatrix} &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \\ &1\ &1\ &1\ &1\ \end{pmatrix}}\\ \end{aligned}$$
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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.