Matrix Multiplication

Table of Contents

The multiplication of matrices means rows of matrix is multiplied to columns of to obtain a third matrix . 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 and are two matrices with sizes and respectively. The following conditions apply to matrix multiplication,

Matrix Multiplication

Let and 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, has size where and has size where the resultant matrix has size of which is .

Now, we consider the second case, where we multiply to .

\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 is where and size of matrix is also where . Therefore, resultant matrix after multiplication has a size of where . In other words, .

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 is where and the size of the matrix is where . Therefore, the resultant matrix has a size of which is .

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 has a size of where and matrix has a size of where . The size of resultant matrix is which is .

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 and ,

\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 is not equal to the product of . Hence, the commutative law does not work in the case of matrix multiplication.