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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\ \\ &2\ \\ &3\ \\ &4\ \\ &5\ \\ &6\ \\ &7\ \\ &8\ \\ &9\ \\ &10\ \\ &11\ \\ &12\ \\ &13\ \\ &14\ \\ &15\ \\ &16\ \end{pmatrix}\times \begin{pmatrix} &1\ &2\ &3\ &4\ &5\ &6\ &7\ &8\ &9\ &10\ &11\ &12\ &13\ &14\ &15\ &16\ \end{pmatrix}}\\ \\Solution:&\\&\begin{pmatrix} &1\ \\ &2\ \\ &3\ \\ &4\ \\ &5\ \\ &6\ \\ &7\ \\ &8\ \\ &9\ \\ &10\ \\ &11\ \\ &12\ \\ &13\ \\ &14\ \\ &15\ \\ &16\ \end{pmatrix}\times \begin{pmatrix} &1\ &2\ &3\ &4\ &5\ &6\ &7\ &8\ &9\ &10\ &11\ &12\ &13\ &14\ &15\ &16\ \end{pmatrix}\\\\=\ \ &\begin{pmatrix} &1\ &2\ &3\ &4\ &5\ &6\ &7\ &8\ &9\ &10\ &11\ &12\ &13\ &14\ &15\ &16\ \\ &2\ &4\ &6\ &8\ &10\ &12\ &14\ &16\ &18\ &20\ &22\ &24\ &26\ &28\ &30\ &32\ \\ &3\ &6\ &9\ &12\ &15\ &18\ &21\ &24\ &27\ &30\ &33\ &36\ &39\ &42\ &45\ &48\ \\ &4\ &8\ &12\ &16\ &20\ &24\ &28\ &32\ &36\ &40\ &44\ &48\ &52\ &56\ &60\ &64\ \\ &5\ &10\ &15\ &20\ &25\ &30\ &35\ &40\ &45\ &50\ &55\ &60\ &65\ &70\ &75\ &80\ \\ &6\ &12\ &18\ &24\ &30\ &36\ &42\ &48\ &54\ &60\ &66\ &72\ &78\ &84\ &90\ &96\ \\ &7\ &14\ &21\ &28\ &35\ &42\ &49\ &56\ &63\ &70\ &77\ &84\ &91\ &98\ &105\ &112\ \\ &8\ &16\ &24\ &32\ &40\ &48\ &56\ &64\ &72\ &80\ &88\ &96\ &104\ &112\ &120\ &128\ \\ &9\ &18\ &27\ &36\ &45\ &54\ &63\ &72\ &81\ &90\ &99\ &108\ &117\ &126\ &135\ &144\ \\ &10\ &20\ &30\ &40\ &50\ &60\ &70\ &80\ &90\ &100\ &110\ &120\ &130\ &140\ &150\ &160\ \\ &11\ &22\ &33\ &44\ &55\ &66\ &77\ &88\ &99\ &110\ &121\ &132\ &143\ &154\ &165\ &176\ \\ &12\ &24\ &36\ &48\ &60\ &72\ &84\ &96\ &108\ &120\ &132\ &144\ &156\ &168\ &180\ &192\ \\ &13\ &26\ &39\ &52\ &65\ &78\ &91\ &104\ &117\ &130\ &143\ &156\ &169\ &182\ &195\ &208\ \\ &14\ &28\ &42\ &56\ &70\ &84\ &98\ &112\ &126\ &140\ &154\ &168\ &182\ &196\ &210\ &224\ \\ &15\ &30\ &45\ &60\ &75\ &90\ &105\ &120\ &135\ &150\ &165\ &180\ &195\ &210\ &225\ &240\ \\ &16\ &32\ &48\ &64\ &80\ &96\ &112\ &128\ &144\ &160\ &176\ &192\ &208\ &224\ &240\ &256\ \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.