# Multiplication of Matrices

A mathematical approach of multiplying a matrix with another matrix is called multiplication of matrices.

Matrices are involved in multiplication in some cases. The product of them can be obtained in matrix form by multiplying elements in a row of one matrix with the corresponding elements in a column of another matrix. The products of elements of a row and a column are added for the elements of product matrix.

Remember, the multiplication of matrices can only be performed when the number of elements in a row is equal to number of elements in a column.

## Example

$A =$ $\begin{bmatrix} 2 & 3 \\ 5 & 4 \\ \end{bmatrix}$ and $B =$ $\begin{bmatrix} 3 & 4 \\ 9 & 1 \\ \end{bmatrix}$ are two matrices of order $2$.

Now, multiply the matrices $A$ and $B$ and it is written as $A \times B$ but the product of them is simply written as $AB$ in mathematics.

$\implies A \times B$ $\,=\,$ $\begin{bmatrix} 2 & 3 \\ 5 & 4 \\ \end{bmatrix}$ $\times$ $\begin{bmatrix} 3 & 4 \\ 9 & 1 \\ \end{bmatrix}$

$\implies AB$ $\,=\,$ $\begin{bmatrix} 2 & 3 \\ 5 & 4 \\ \end{bmatrix}$ $\times$ $\begin{bmatrix} 3 & 4 \\ 9 & 1 \\ \end{bmatrix}$

Multiply elements of each row with elements of each column and then add product of them in every step. It will be an element of product matrix.

The matrix $A$ has two rows and matrix $B$ has two columns in this example. Firstly, multiply the elements of first row of matrix $A$ with the elements of first column of matrix $B$.

$\implies AB$ $\,=\,$ $\begin{bmatrix} 2 \times 3 + 3 \times 9 & \\ & \\ \end{bmatrix}$

$\implies AB$ $\,=\,$ $\begin{bmatrix} 33 & \\ & \\ \end{bmatrix}$

Now, multiply the elements of first row of matrix $A$ with elements of second column of matrix $B$, and then add products of them for the element of first row, second column of the product matrix.

$\implies AB$ $\,=\,$ $\begin{bmatrix} 33 & 2 \times 4 + 3 \times 1 \\ & \\ \end{bmatrix}$

$\implies AB$ $\,=\,$ $\begin{bmatrix} 33 & 11 \\ & \\ \end{bmatrix}$

There is no columns in matrix $B$ to multiply by the first row of matrix $A$. So, multiply elements of second row of matrix $A$ with elements of both columns of matrix $B$.

$\implies AB$ $\,=\,$ $\begin{bmatrix} 33 & 11 \\ 5 \times 3 + 4 \times 9 & 5 \times 4 + 4 \times 1 \\ \end{bmatrix}$

$\,\,\, \therefore \,\,\,\,\,\, AB$ $\,=\,$ $\begin{bmatrix} 33 & 11 \\ 51 & 24 \\ \end{bmatrix}$

Thus, the multiplication of matrices is performed in matrix to obtain product of them in matrix form.

The matrices $A$ and $B$ are of same order. So, there is no difficulty in performing the multiplication. Even, the multiplication can be performed with different order matrices but it is failed in some cases of different order matrices. It is possible when the number of elements in a row of one matrix is not equal to number of elements in a column of another matrix.

$C =$ $\begin{bmatrix} 1 & 5 & -2 \\ 0 & 2 & 3\\ \end{bmatrix}$ and $D =$ $\begin{bmatrix} 4 & 2 \\ 7 & 9 \\ \end{bmatrix}$ are two matrices.

The multiplication of the matrices $C$ and $D$ is written as $C \times D$ and the product of them is written as $CD$.

$\implies C \times D$ $=$ $\begin{bmatrix} 1 & 5 & -2 \\ 0 & 2 & 3\\ \end{bmatrix}$ $\times$ $\begin{bmatrix} 4 & 2 \\ 7 & 9 \\ \end{bmatrix}$

The first row of matrix $C$ has $3$ elements and the first column of matrix $D$ has $2$ elements. So, it is not possible to multiply them and it is not possible to get the element of product matrix. Therefore, the multiplication of matrices $C$ and $D$ is failed and it is not possible.

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