If $A$ is an $n\times m $ matrix and $B$ is an $m \times p$ matrix, the result $AB$ of their multiplication is an $n \times p$ matrix defined only if the number of columns $m$ in $A$ is equal to the number of rows $m$ in $B$.  Check to make sure that this is true before multiplying the matrices, since there is "no solution" otherwise.

General Definition and Process: Matrix Multiplication

Scalar multiplication is simply multiplying a value through all the elements of a matrix, whereas matrix multiplication is multiplying every element of each row of the first matrix times every element of each column in the second matrix.  Scalar multiplication is much more simple than matrix multiplication; however, a pattern does exist.    

When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix. Each entry of the resultant matrix is computed one at a time. 

For two matrices the final position of the product is shown below:

$\displaystyle \begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } \\ \cdot & \cdot \\ { a }_{ 31 } & { a }_{ 32 } \\ \cdot & \cdot \end{bmatrix}\begin{bmatrix} \cdot & { b }_{ 12 } & { b }_{ 13 } \\ \cdot & { b }_{ 22 } & { b }_{ 23 } \end{bmatrix}=\begin{bmatrix} \cdot & x_{ 12 } & \cdot \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & { x }_{ 33 } \\ \cdot & \cdot & \cdot \end{bmatrix}$

Matrix Multiplication

This figure illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B.

The values at the intersections marked with circles are:

$\displaystyle {x}_{12}=({a}_{11},{a}_{12}) \cdot ({b}_{12},{b}_{22})=({a}_{11} {b}_{12}) +({a}_{12} {b}_{22})$

$\displaystyle {x}_{33}=({a}_{31},{a}_{32}) \cdot ({b}_{13},{b}_{23})=({a}_{31} {b}_{13}) +({a}_{32} {b}_{23})$

Matrix Multiplication: Process

Example 1:  Find the product $AB$

$\displaystyle A=\begin{pmatrix} { 1 } & { 2 } \\ { 3 } & { 4 } \end{pmatrix}\quad B=\begin{pmatrix} { 5 } & { 6 } \\ { 7 } & { 8 } \end{pmatrix} $

First ask: Do the number of columns in $A$ equal the number of rows in $B$?  The number of columns in $A$ is $2$, and the number of rows in $B$ is also $2$, therefore a product exists.

Start with producing the product for the first row, first column element.  Take the first row of Matrix $A$ and multiply by the first column of Matrix $B$:  The first element of $A$ times the first column element of $B$, plus the second element of $A$ times the second column element of $B$.

$\displaystyle AB=\begin{pmatrix} { (1 \cdot 5) }+{ (2 \cdot 7) } & ({ })+{ ( )} \\ { ( ) }+{ ( ) } & { ( ) }+{ ( ) } \end{pmatrix}$

Continue the pattern with the first row of $A$ by the second column of $B$, and then repeat with the second row of $A$.

AB has entries defined by the equation:

$\displaystyle AB=\begin{pmatrix} { (1 \cdot 5) }+{ (2 \cdot 7) } & ({ 1 \cdot 6})+{ (2 \cdot 8)} \\ { (3 \cdot 5) }+{ (4 \cdot 7) } & { (3 \cdot 6) }+{ (4 \cdot 8) } \end{pmatrix}$

$\displaystyle AB=\begin{pmatrix} {(5+14)} & {(6+16)} \\ {(15+28)} & {(18+32)} \end{pmatrix}$

$\displaystyle AB= \begin{pmatrix} {(19)} & {(22)} \\ {(43)} & {(50)} \end{pmatrix}$