Matrix Multiplication

Table of Contents

The multiplication of matrices means rows of matrix $A$ is multiplied to columns of $B$ to obtain a third matrix $C$ . We also evaluate the matrix multiplication with respect to fundamental properties of mathematics such as commutative, associative property, identity property.

Conditions for Matrix Multiplication

If $A_{m \times n}$ and $B_{n \times p}$ are two matrices with sizes $m \times n$ and $n \times p$ respectively. The following conditions apply to matrix multiplication,

Row or column of matrix $A$ must be equal to column or row of matrix $B$ .
Multiplying matrix $A$ to matrix $B$ is not same as multiplying matrix $B$ i.e., $AB \neq BA$ .
If condition 2 and 3 are true, then we can multiply 2 or more matrices.

Matrix Multiplication

Let $A_{1 \times n}$ and $B_{n \times 1}$ be two matrices with elements. Then the matrix multiplication can be done as follows.

\begin{aligned} &A = \begin{bmatrix}a_{11} & a_{12}& . . . & a_{1n}\end{bmatrix} B = \begin{bmatrix}b_{11} \\ b_{21} \\ : \\ b_{n1}\end{bmatrix}\\\\ &C_{1 \times 1} = A . B = \begin{bmatrix}a_{11}b_{11} + a_{12}b_{21} + ... + a_{1n}b_{n1}\end{bmatrix} \end{aligned}

In the above example, $A$ has size $m \times n$ where $m = 1$ and $B$ has size $n \times p$ where the resultant matrix $C$ has size of $m \times p$ which is $1 \times 1$ .

Now, we consider the second case, where we multiply $B$ to $A$ .

\begin{aligned} &B = \begin{bmatrix}b_{11} \\ b_{21} \\ : \\ b_{n1}\end{bmatrix} A = \begin{bmatrix}a_{11} & a_{12}& . . . & a_{1n}\end{bmatrix}\\\\ &C_{n \times n} = A . B = \begin{bmatrix}b_{11}a_{11} & b_{11}a_{12} & ... & b_{11}a_{1n}\\ b_{21}a_{11} & b_{21}a_{11} & ... & b_{21}a_{1n}\\ : & : & : & :\\ b_{n1}a_{11} & b_{n1}a_{12} & ... & b_{n1}a_{1n}\end{bmatrix} \end{aligned}

Matrix Multiplication Examples

In this section, we will show you few examples with different kinds of matrices.

Example #1

\begin{aligned} &// \hspace{5px} Multiplying \hspace{5px} Square \hspace{5px}Matrix\\\\ &A_{2 \times 2} = \begin{bmatrix}2 & 3 \\ 1 & 7\end{bmatrix} B_{2 \times 2} = \begin{bmatrix}2 & 3\\ 1 & 1\end{bmatrix}\\\\ &AB = \begin{bmatrix}(2 * 2)+ (3 * 1) & (2 * 3)+ (3 * 1)\\ (1 * 2) + (7 * 1) & (1 * 3) + ( 7 * 1)\end{bmatrix}\\\\ &AB = \begin{bmatrix}(4)+ (3) & (6)+ (3)\\ (2) + (7) & (3) + (7)\end{bmatrix}\\\\ &AB_{2 \times 2} = \begin{bmatrix}7 & 9\\ 9 & 10\end{bmatrix} \end{aligned}

The size of matrix $A$ is $m \times n$ where $m = n$ and size of matrix $B$ is also $n \times p$ where $n = p$ . Therefore, resultant matrix after multiplication has a size of $m \times p$ where $m = p$ . In other words, $A_{2 \times 2} \times B_{2 \times 2} = C_{2 \times 2}$ .

Example #2

\begin{aligned} &A_{2 \times 3} = \begin{bmatrix}4 & -1 & 1\\ 4 & 1 & -2\end{bmatrix} B_{3 \times 2} = \begin{bmatrix}2 & -2\\ 1 & -2 \\ 5 & 2\end{bmatrix}\\\\ &AB = \begin{bmatrix}(4 * 2)+ (-1 * 1) + (1 * 5) & (4 * -2)+ (-1 * -2) + (1 * 2)\\ (4 * 2) + (1 * 1) + (-2 * 5) & (4 * -2) + ( 1 * -2) + (-2 * 2)\end{bmatrix}\\\\ &AB = \begin{bmatrix}(8)+ (-1) + (5) & (-8)+ (2) + (2)\\ (8) + (1) + (-10) & (-8) + (-2) + (-4)\end{bmatrix}\\\\ &AB = \begin{bmatrix}12 & -4\\ -1 & -14\end{bmatrix} \end{aligned}

The size of the matrix $A$ is $m \times n$ where $m = 2$ and the size of the matrix $B$ is $n \times p$ where $p = 2$ . Therefore, the resultant matrix $AB$ has a size of $m \times p$ which is $2 \times 2$ .

Example #3

\begin{aligned} &A_{3 \times 2} = \begin{bmatrix}1 & 1 \\ 3 & 5 \\ 2 & 3\end{bmatrix} B_{2 \times 3} = \begin{bmatrix}3 & 1 & 0\\ 7 & 4 & 1 \end{bmatrix}\\\\ &AB = \begin{bmatrix}(1 * 3)+ (1 * 7) & (1 * 1) + (1 * 4) & (1 * 0) + (1 * 1)\\ (3 * 3)+ (5 * 7) & (3 * 1) + (5 * 4) & (3 * 0) + (5 * 1)\\ (2 * 3)+ (3 * 7) & (2 * 1) + (3 * 4) & (2 * 0) + (3 * 1)\\\end{bmatrix}\\\\ &AB = \begin{bmatrix}(3)+ (7) & (1) + (4) & (0) + (1)\\ (9)+ (35) & (3) + (20) & (0) + (5)\\ (6)+ (21) & (2) + (12) & (0) + (3)\\\end{bmatrix}\\\\ &AB = \begin{bmatrix}10 & 4 & 1\\ 44 & 23 & 5\\ 27 & 14 & 3\end{bmatrix} \end{aligned}

The matrix $A$ has a size of $m \times n$ where $m = 3$ and matrix $B$ has a size of $n \times p$ where $p = 3$ . The size of resultant matrix $AB$ is $m \times p$ which is $3 \times 3$ .

Properties of Matrix Multiplication

Since, matrix operation is mathematical operations, therefore, matrix multiplication must preserve all or some of the properties with respect to multiplication operator.

Commutative Law

Suppose we multiply two matrix $A$ and $B$ ,

\begin{aligned} &A = \begin{bmatrix}2 & 2\\ 3 & 1\end{bmatrix} B = \begin{bmatrix}1 & 2\\ 5 & 2\end{bmatrix}\\\\ &AB = \begin{bmatrix}12 & 8\\ 8 & 8\end{bmatrix}\\\\ &Similarly, \\\\ &B = \begin{bmatrix}1 & 2\\ 5 & 2\end{bmatrix} A = \begin{bmatrix}2 & 2\\ 3 & 1\end{bmatrix}\\\\ &BA = AB = \begin{bmatrix}8 & 4\\ 16 & 12\end{bmatrix}\\\\ &AB \neq BA \end{aligned}

From the example above, it is clear that the product of $AB$ is not equal to the product of $BA$ . Hence, the commutative law does not work in the case of matrix multiplication.

Associative Law of Matrix Multiplication

Conditions for Matrix Multiplication

Matrix Multiplication

Matrix Multiplication Examples

Properties of Matrix Multiplication

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