Mathematics · Ch 3 — Matrices

Multiplication of Matrices

3.4.5

Multiplication of Matrices

3.4.5 Multiplication of Matrices

Understanding the Need for Matrix Multiplication

Meera wants 2 pens and 5 story books; Nadeem needs 8 pens and 10 story books. At the first shop a pen costs ₹5 and a story book ₹50, so Meera needs $5 \times 2 + 50 \times 5 = 260$ rupees and Nadeem $8 \times 5 + 10 \times 50 = 540$ rupees.

The requirements form a $2 \times 2$ matrix and the prices a $2 \times 1$ matrix; multiplying each row of requirements by the price column gives the money needed:

\begin{bmatrix} 2 & 5 \\ 8 & 10 \end{bmatrix}\begin{bmatrix} 5 \\ 50 \end{bmatrix} = \begin{bmatrix} 5 \times 2 + 50 \times 5 \\ 8 \times 5 + 10 \times 50 \end{bmatrix} = \begin{bmatrix} 260 \\ 540 \end{bmatrix}

Now suppose they check another shop where pens cost ₹4 and story books ₹40. Then Meera needs $4 \times 2 + 40 \times 5 = 208$ rupees and Nadeem $8 \times 4 + 10 \times 40 = 432$ rupees. Combining both price scenarios into one $2 \times 2$ price matrix:

\begin{bmatrix} 2 & 5 \\ 8 & 10 \end{bmatrix} \begin{bmatrix} 5 & 4 \\ 50 & 40 \end{bmatrix} = \begin{bmatrix} 5 \times 2 + 50 \times 5 & 4 \times 2 + 40 \times 5 \\ 8 \times 5 + 10 \times 50 & 8 \times 4 + 10 \times 40 \end{bmatrix} = \begin{bmatrix} 260 & 208 \\ 540 & 432 \end{bmatrix}

The idea: take rows from the first matrix and columns from the second, multiply corresponding entries, and add them up.

Important

For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix.

Formal Definition of Matrix Multiplication

Let $A = [a_{ij}]$ be an $m \times n$ matrix and $B = [b_{jk}]$ an $n \times p$ matrix. The product $AB$ is a matrix $C = [c_{ik}]$ of order $m \times p$ . To find the element $c_{ik}$ (in the $i$ -th row and $k$ -th column of the product), take the $i$ -th row of $A$ ,

[a_{i1}\; a_{i2}\; \dots\; a_{in}],

and the $k$ -th column of $B$ ,

\begin{bmatrix} b_{1k} \\ b_{2k} \\ \vdots \\ b_{nk} \end{bmatrix},

multiply corresponding elements, and sum them:

c_{ik} = a_{i1}b_{1k} + a_{i2}b_{2k} + a_{i3}b_{3k} + \dots + a_{in}b_{nk} = \sum_{j=1}^{n} a_{ij}b_{jk}.

The index $j$ runs from 1 to $n$ , matching the columns of $A$ with the rows of $B$ .

Worked Example: Computing a Product

Let

C = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 3 & 4 \end{bmatrix}_{2 \times 3}, \quad D = \begin{bmatrix} 2 & 7 \\ -1 & 1 \\ 5 & -4 \end{bmatrix}_{3 \times 2}

Since $C$ has 3 columns and $D$ has 3 rows, the product $CD$ is defined and is a $2 \times 2$ matrix. Computing each entry as (row of $C$ ) · (column of $D$ ):

\begin{aligned} c_{11} &= 1(2) + (-1)(-1) + 2(5) = 2 + 1 + 10 = 13, \\ c_{12} &= 1(7) + (-1)(1) + 2(-4) = 7 - 1 - 8 = -2, \\ c_{21} &= 0(2) + 3(-1) + 4(5) = 0 - 3 + 20 = 17, \\ c_{22} &= 0(7) + 3(1) + 4(-4) = 0 + 3 - 16 = -13. \end{aligned}

Therefore

CD = \begin{bmatrix} 13 & -2 \\ 17 & -13 \end{bmatrix}.

Conditions for Both Products to Exist

If $A$ is $m \times n$ and $B$ is $k \times l$ , then $AB$ is defined iff $n = k$ , and $BA$ is defined iff $l = m$ . Both $AB$ and $BA$ are defined iff $n = k$ and $l = m$ . In particular, if $A$ and $B$ are square of the same order $n \times n$ , both products are defined and are $n \times n$ .

Non-Commutativity of Matrix Multiplication

Even when both $AB$ and $BA$ are defined, they are not necessarily equal — matrix multiplication is not commutative. …

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