Matrix Multiplication

Understanding the Multiplication of Two Matrices

Matrix multiplication is a fundamental operation in linear algebra. Unlike matrix addition or subtraction where elements are operated on directly, matrix multiplication has its own specific rules.

Condition for Matrix Multiplication

Two matrices, let's say matrix $A$ and matrix $B$, can be multiplied ($A \times B$) only if the number of columns in matrix $A$ is equal to the number of rows in matrix $B$.

Suppose matrix $A$ has an order of $m \times n$ (meaning $m$ rows and $n$ columns) and matrix $B$ has an order of $n \times p$ (meaning $n$ rows and $p$ columns).

Since the number of columns in matrix $A$ ($n$) is equal to the number of rows in matrix $B$ ($n$), then matrix $A$ and $B$ can be multiplied.

The result of the multiplication, let's call it matrix $C = AB$, will have an order of $m \times p$.

How to Calculate the Elements of the Resultant Matrix

The element $c_{ij}$ in matrix $C$ (i.e., the element in the $i$-th row and $j$-th column) is calculated by multiplying each element in the $i$-th row of matrix $A$ by the corresponding element in the $j$-th column of matrix $B$, and then summing all these products.

Mathematically, if $A = [a_{ik}]$ and $B = [b_{kj}]$, then the element $c_{ij}$ of matrix $C = AB$ is:

The notation $\sum$ (sigma) means summation.

In the formula above, we sum the products $a_{ik}b_{kj}$ for all values of $k$ from 1 to $n$.

Steps to Multiply Matrices

Let's look at a simple example to understand the process.

Suppose we have matrices $P = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix}$ and $Q = \begin{bmatrix} q_{11} & q_{12} \\ q_{21} & q_{22} \end{bmatrix}$.

Matrix $P$ has an order of $2 \times 2$ and matrix $Q$ also has an order of $2 \times 2$. The number of columns in $P$ (which is 2) is equal to the number of rows in $Q$ (which is 2), so we can multiply them. The result, $R = PQ$, will have an order of $2 \times 2$.

The elements of matrix $R$ are calculated as follows:

Example of Multiplying Two Matrices

Given two matrices:

Matrix $A$ has an order of $3 \times 3$ and matrix $B$ has an order of $3 \times 2$.

The number of columns in matrix $A$ (which is 3) is equal to the number of rows in matrix $B$ (which is 3).

So, $AB$ can be calculated and will result in a matrix of order $3 \times 2$.

Let's calculate $C = AB$:

So, the result of the matrix multiplication $AB$ is:

Now, what about $BA$?

Matrix $B$ has an order of $3 \times 2$ and matrix $A$ has an order of $3 \times 3$.

The number of columns in matrix $B$ (which is 2) is not equal to the number of rows in matrix $A$ (which is 3).

Therefore, the multiplication $BA$ is undefined. This illustrates one of the important properties of matrix multiplication.

Properties of Matrix Multiplication

Matrix multiplication has several important properties:

1. Generally Not Commutative:

   This means $AB \neq BA$. We have already seen an example above where $AB$ is defined but $BA$ is not. Even if both are defined, the results are not necessarily the same.

2. Associative:

   If the matrix multiplications $A, B,$ and $C$ are defined, then $(AB)C = A(BC)$ holds. This means the order of grouping the multiplication does not change the final result.

3. Distributive:

   Matrix multiplication is distributive over matrix addition or subtraction:

   $$A(B + C) = AB + AC$$

   $$(A + B)C = AC + BC$$

   This holds if all involved addition and multiplication operations are defined.

4. Multiplication by Identity Matrix ($I$):

   If $A$ is a square matrix of order $n \times n$ and $I$ is the identity matrix of order $n \times n$, then:

   $$AI = IA = A$$

   The identity matrix acts like the number 1 in ordinary number multiplication.

5. Multiplication by a Scalar ($k$):

   If $k$ is a scalar (real number), then:

   $$k(AB) = (kA)B = A(kB)$$

Calculating Revenue

Matrix multiplication is very useful in various fields, one of which is for managing data and calculating aggregate values.

Imagine a home industry produces three types of food: tempeh chips, banana chips, and potato chips.

These foods are marketed in three places: Place A, Place B, and Place C.

The number of chips (in jars) sold in each place is presented in matrix $P$. The columns in matrix $P$ respectively represent Place A, Place B, and Place C, while the rows respectively represent tempeh chips, banana chips, and potato chips.

The first row ($\begin{bmatrix} 15 & 12 & 20 \end{bmatrix}$) means 15 jars of tempeh chips were sold in Place A, 12 in Place B, and 20 in Place C.

The price for each jar of chips (in rupiah) is stated in the column matrix $Q$ below:

To determine the total revenue from each type of chip across all places, we can multiply matrix $P$ by matrix $Q$.

However, pay attention to the order of the matrices. Matrix $P$ has an order of $3 \times 3$ and matrix $Q$ has an order of $3 \times 1$. The number of columns in $P$ (3) is equal to the number of rows in $Q$ (3), so $PQ$ can be calculated and will result in a matrix $R$ of order $3 \times 1$.

Matrix $R = PQ$ will show the total revenue for each type of chip.

From matrix $R$, we can see that the total revenue from the sale of tempeh chips is Rp1,080,000, banana chips Rp1,100,000, and potato chips Rp1,125,000.

If the question is "determine the revenue matrix for each place", then we need to arrange the price matrix $Q$ differently or perform multiplication with the transpose of $P$.

Suppose we want to find the total revenue in Place A, Place B, and Place C. We can use the price matrix as a row matrix $Q^T = \begin{bmatrix} 20,000 & 15,000 & 30,000 \end{bmatrix}$ and multiply it by matrix $P$: $S = Q^T P$.

Matrix $Q^T$ has an order of $1 \times 3$ and $P$ has an order of $3 \times 3$. The result $S$ will have an order of $1 \times 3$.

Thus, $S = \begin{bmatrix} 1,125,000 & 840,000 & 1,225,000 \end{bmatrix}$.

This means the total revenue from Place A is Rp1,125,000, from Place B is Rp840,000, and from Place C is Rp1,225,000.

The interpretation of the elements of the resulting matrix greatly depends on how the initial matrices are defined and how the multiplication is performed.

Exercises

Given the following matrices:

Determine the matrices $CD$ and $DC$.

Is $CD = DC$?

Answer Key

1. Calculating $CD$:

   Matrix $C$ has an order of $2 \times 2$ and $D$ has an order of $2 \times 2$. The result will have an order of $2 \times 2$.

   $$CD = \begin{bmatrix} 1 & -3 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 1 & -4 \\ 8 & 2 \end{bmatrix}$$

   $$CD = \begin{bmatrix} (1)(1) + (-3)(8) & (1)(-4) + (-3)(2) \\ (1)(1) + (2)(8) & (1)(-4) + (2)(2) \end{bmatrix}$$

   $$CD = \begin{bmatrix} 1 - 24 & -4 - 6 \\ 1 + 16 & -4 + 4 \end{bmatrix}$$

   $$CD = \begin{bmatrix} -23 & -10 \\ 17 & 0 \end{bmatrix}$$

2. Calculating $DC$:

   Matrix $D$ has an order of $2 \times 2$ and $C$ has an order of $2 \times 2$. The result will have an order of $2 \times 2$.

   $$DC = \begin{bmatrix} 1 & -4 \\ 8 & 2 \end{bmatrix} \begin{bmatrix} 1 & -3 \\ 1 & 2 \end{bmatrix}$$

   $$DC = \begin{bmatrix} (1)(1) + (-4)(1) & (1)(-3) + (-4)(2) \\ (8)(1) + (2)(1) & (8)(-3) + (2)(2) \end{bmatrix}$$

   $$DC = \begin{bmatrix} 1 - 4 & -3 - 8 \\ 8 + 2 & -24 + 4 \end{bmatrix}$$

   $$DC = \begin{bmatrix} -3 & -11 \\ 10 & -20 \end{bmatrix}$$

From the results above, it is clear that $CD = \begin{bmatrix} -23 & -10 \\ 17 & 0 \end{bmatrix}$ and $DC = \begin{bmatrix} -3 & -11 \\ 10 & -20 \end{bmatrix}$.

So, $CD \neq DC$. This is another example showing that matrix multiplication is generally not commutative.