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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
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &3\ &-1\ \\ &-2\ &1\ \end{pmatrix}\times \begin{pmatrix} &-1\ &1\ \\ &0\ &1\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{853}{500}\ &\frac{231}{200}\ &\frac{259}{500}\ &-\frac{1707}{1000}\ &\frac{1787}{1000}\ \\ &-\frac{19}{100}\ &-\frac{1133}{1000}\ &-\frac{207}{500}\ &\frac{37}{250}\ &-\frac{523}{500}\ \\ &-\frac{133}{200}\ &\frac{207}{250}\ &-\frac{207}{1000}\ &\frac{153}{250}\ &\frac{153}{1000}\ \\ &-\frac{1403}{1000}\ &-\frac{621}{500}\ &-\frac{29}{20}\ &-\frac{63}{200}\ &-\frac{13}{200}\ \\ &\frac{591}{500}\ &\frac{49}{125}\ &\frac{777}{500}\ &\frac{1261}{1000}\ &-\frac{207}{250}\ \end{pmatrix}\times \begin{pmatrix} &\frac{337}{1000}\ &-\frac{9}{200}\ &\frac{457}{1000}\ &\frac{273}{1000}\ &-\frac{9}{125}\ \\ &-\frac{34}{125}\ &-\frac{213}{500}\ &-\frac{24}{125}\ &-\frac{17}{200}\ &\frac{61}{250}\ \\ &-\frac{163}{1000}\ &-\frac{9}{250}\ &-\frac{409}{1000}\ &-\frac{457}{1000}\ &\frac{3}{40}\ \\ &-\frac{62}{125}\ &\frac{723}{1000}\ &-\frac{31}{125}\ &\frac{169}{500}\ &\frac{2}{25}\ \\ &\frac{21}{200}\ &\frac{7}{100}\ &-\frac{97}{1000}\ &-\frac{129}{250}\ &\frac{46}{125}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{269}{250}\ &\frac{231}{200}\ &\frac{259}{500}\ &-\frac{1707}{1000}\ &\frac{1787}{1000}\ \\ &-\frac{19}{100}\ &-\frac{1133}{1000}\ &-\frac{207}{500}\ &\frac{37}{250}\ &-\frac{523}{500}\ \\ &-\frac{133}{200}\ &\frac{207}{250}\ &-\frac{207}{1000}\ &\frac{153}{250}\ &\frac{153}{1000}\ \\ &-\frac{1403}{1000}\ &-\frac{621}{500}\ &-\frac{29}{20}\ &-\frac{63}{200}\ &-\frac{13}{200}\ \\ &\frac{591}{500}\ &\frac{49}{125}\ &\frac{777}{500}\ &\frac{1261}{1000}\ &-\frac{207}{250}\ \end{pmatrix}\times \begin{pmatrix} &-\frac{83}{1000}\ &-\frac{333}{1000}\ &\frac{74}{125}\ &-\frac{93}{250}\ &\frac{373}{200}\ \\ &-\frac{247}{500}\ &-\frac{87}{250}\ &\frac{319}{500}\ &-\frac{457}{1000}\ &-\frac{447}{1000}\ \\ &-\frac{509}{1000}\ &-\frac{573}{1000}\ &\frac{53}{500}\ &\frac{248}{125}\ &-\frac{39}{125}\ \\ &-\frac{427}{500}\ &\frac{1979}{1000}\ &\frac{5}{8}\ &-\frac{423}{1000}\ &\frac{1}{1000}\ \\ &\frac{97}{50}\ &-\frac{181}{250}\ &-\frac{49}{25}\ &-\frac{733}{1000}\ &-\frac{1107}{1000}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &-1\ &-8192\ \\ &-1\ &2048\ \end{pmatrix}\times \begin{pmatrix} &1\ &4\ \\ &-1\ &-1\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &-4\ \\ &1\ &1\ \end{pmatrix}\times \begin{pmatrix} &-1\ &0\ \\ &0\ &2048\ \end{pmatrix}}\\ \end{aligned}$$
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$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{853}{500}\ &\frac{231}{200}\ &\frac{259}{500}\ &-\frac{1707}{1000}\ &\frac{1787}{1000}\ \\ &-\frac{19}{100}\ &-\frac{1133}{1000}\ &-\frac{207}{500}\ &\frac{37}{250}\ &-\frac{523}{500}\ \\ &-\frac{133}{200}\ &\frac{207}{250}\ &-\frac{207}{1000}\ &\frac{153}{250}\ &\frac{153}{1000}\ \\ &-\frac{1403}{1000}\ &-\frac{621}{500}\ &-\frac{29}{20}\ &-\frac{63}{200}\ &-\frac{13}{200}\ \\ &\frac{591}{500}\ &\frac{49}{125}\ &\frac{777}{500}\ &\frac{1261}{1000}\ &-\frac{207}{250}\ \end{pmatrix}\times \begin{pmatrix} &\frac{337}{1000}\ &-\frac{9}{200}\ &\frac{457}{1000}\ &\frac{273}{1000}\ &-\frac{9}{125}\ \\ &-\frac{34}{125}\ &-\frac{213}{500}\ &-\frac{24}{125}\ &-\frac{17}{200}\ &\frac{61}{250}\ \\ &-\frac{163}{1000}\ &-\frac{9}{250}\ &-\frac{409}{1000}\ &-\frac{457}{1000}\ &\frac{3}{40}\ \\ &-\frac{62}{125}\ &\frac{723}{1000}\ &-\frac{31}{125}\ &\frac{169}{500}\ &\frac{2}{25}\ \\ &\frac{21}{200}\ &\frac{7}{100}\ &-\frac{97}{1000}\ &-\frac{129}{250}\ &\frac{46}{125}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &\frac{269}{250}\ &\frac{231}{200}\ &\frac{259}{500}\ &-\frac{1707}{1000}\ &\frac{1787}{1000}\ \\ &-\frac{19}{100}\ &-\frac{1133}{1000}\ &-\frac{207}{500}\ &\frac{37}{250}\ &-\frac{523}{500}\ \\ &-\frac{133}{200}\ &\frac{207}{250}\ &-\frac{207}{1000}\ &\frac{153}{250}\ &\frac{153}{1000}\ \\ &-\frac{1403}{1000}\ &-\frac{621}{500}\ &-\frac{29}{20}\ &-\frac{63}{200}\ &-\frac{13}{200}\ \\ &\frac{591}{500}\ &\frac{49}{125}\ &\frac{777}{500}\ &\frac{1261}{1000}\ &-\frac{207}{250}\ \end{pmatrix}\times \begin{pmatrix} &-\frac{83}{1000}\ &-\frac{333}{1000}\ &\frac{74}{125}\ &-\frac{93}{250}\ &\frac{373}{200}\ \\ &-\frac{247}{500}\ &-\frac{87}{250}\ &\frac{319}{500}\ &-\frac{457}{1000}\ &-\frac{447}{1000}\ \\ &-\frac{509}{1000}\ &-\frac{573}{1000}\ &\frac{53}{500}\ &\frac{248}{125}\ &-\frac{39}{125}\ \\ &-\frac{427}{500}\ &\frac{1979}{1000}\ &\frac{5}{8}\ &-\frac{423}{1000}\ &\frac{1}{1000}\ \\ &\frac{97}{50}\ &-\frac{181}{250}\ &-\frac{49}{25}\ &-\frac{733}{1000}\ &-\frac{1107}{1000}\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &-1\ &-8192\ \\ &-1\ &2048\ \end{pmatrix}\times \begin{pmatrix} &1\ &4\ \\ &-1\ &-1\ \end{pmatrix}}\\ \end{aligned}$$
$$ \begin{aligned}&\\ \color{black}{Calculate }& \color{black}{\ \ \begin{pmatrix} &1\ &-4\ \\ &1\ &1\ \end{pmatrix}\times \begin{pmatrix} &-1\ &0\ \\ &0\ &2048\ \end{pmatrix}}\\ \end{aligned}$$
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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.