Matrices

In this article, we will learn about matrices as laid down in the Secondary 4 Mathematics class. We will touch upon the following areas:

• What are Matrices?
• What is the Order of a Matrix?
• What are Equal Matrices?
• How to Add and Subtract Matrices?
• How to Do Scalar Multiplication?

Introduction To Matrices: Order Of A Matrix

A matrix is a rectangular array of numbers. Each of those numbers inside the matrix is known as an element. And the order of a matrix is simply the number of rows by the number of columns.

Example 1:

Write down the order of the matrix

\(\begin{pmatrix}8 & -1 & 0 & 9\\\ 6 & 0 & -1 & 3\end{pmatrix}\)

Solution:

First, we need to find the number of rows and the number of columns.

So, the order is simply \(2 \times 4\). It must always be \(row × column\). 

Equal Matrices

Two matrices are equal only when:

1. They have the same order.
2. All of their corresponding elements must be equal. So, they must have the same numbers in each corresponding position.

Example 2:

If \(A= \begin{pmatrix}8 & \frac{1}{2}n \end{pmatrix}\) , \(B=\begin{pmatrix}4m & -3 \end{pmatrix}\) and \(A = B\), find the value of \(m\) and \(n\).

Solution:

It is given that \(A\) is a row matrix and \(B=\begin{pmatrix}4m & -3 \end{pmatrix}\) and matrix \(A\) equals matrix \(B\).

This means that \(\begin{pmatrix} 8 & \frac{1}{2}n \end{pmatrix} = \begin{pmatrix} 4m & -3 \end{pmatrix}\).

Remember, equal matrices mean where we have the same order and their corresponding elements are also equal. So, it means that \(8\) must be equal to \(4m\) and \(\frac {1}{2}n\) must be equal to \(-3\).

Addition & Subtraction Of Matrices

When two matrices have the same order, only then we can add or subtract them. In this case, we have two matrices, \(A\) and \(B\), where \(A\) is \(\begin{pmatrix}a & b \\\ c & d \end{pmatrix}\) and \(B\) is \(\begin{pmatrix}w & x \\\ y & z \end{pmatrix}\). Since, they have the same order, we can add them, and when we add them, we get

\(A+B =\begin{pmatrix}a+w & b+x \\\ c+y & d+z \end{pmatrix}\)

The order of the matrix does not change. If a \(2 × 2\) matrix is added to a \(2 × 2\) matrix, it’s still a \(2 × 2\).

The same thing is in subtraction. When we minus the corresponding elements, it becomes

\(A-B =\begin{pmatrix}a-w & b-x \\\ c-y & d-z \end{pmatrix}\)

The order of the matrix does not change here.

Example 3:

Evaluate:

 \(\begin{pmatrix}1 & 5 \\\ 2 & -1 \\\ 0 & 4\end{pmatrix} + \begin{pmatrix}-3 & -1 \\\ 0 & 1 \\\ 7 & 5\end{pmatrix}\)

Solution:

When we add them together, we compare the corresponding elements.

\(\begin{pmatrix}1+(-3) & 5+(-1) \\\ 2+0 & -1+1 \\\ 0+7 & 4+5\end{pmatrix}\)

Then, simplify

\(\begin{pmatrix}-2 & 4 \\\ 2 & 0 \\\ 7 & 9\end{pmatrix}\)

This is your final matrix.

Example 4:

Given:

 \(\begin{pmatrix} 3w & 4 & -x \\\ -1 & 0 & 3x \end{pmatrix} - \begin{pmatrix} w & x & 2y \\\ -1 & 5y & -w \end{pmatrix} = \begin{pmatrix} 4 & 2w & 10 \\\ 0 & z & 2 \end{pmatrix}\)

Solution:

Let’s try the left-hand side first. On the left-hand side, we have a matrix subtraction.

\(\begin{pmatrix} 3w-w & 4-x & -x-2y \\\ -1-(-1) & 0-5y & 3x-(-w) \end{pmatrix} = \begin{pmatrix} 4 & 2w & 10 \\\ 0 & z & 2 \end{pmatrix}\)

From here, we simplify,

\(\begin{pmatrix} 2w & 4-x & -x-2y \\\ 0 & -5y & 3x+w) \end{pmatrix} = \begin{pmatrix} 4 & 2w & 10 \\\ 0 & z & 2 \end{pmatrix}\)

We know that these two matrices are equal. So, we can use our knowledge of equal matrices and compare the corresponding elements.

So, first we find \(w\), 

\(\begin{aligned} 2w &= 4; \\   w &= 2\\ \end{aligned}\)

Then, we solve for \(x\). 

So, we know that 

\(\begin{aligned} 4 - x &= 2w\\ 4 - x &= 2 × 2\\ 4 - x &= 4\\ -x &= 0, \\ x &= 0\\ \end{aligned}\)

Next, find \(y\). 

So,

\(- x - 2y = 10\)

Because, \(x = 0\)

So,

\(\begin{aligned} - 0 - 2y &= 10\\ - 2y &= 10\\ y  &= -5\\ \end{aligned}\)

So, when we finally look for the value of \(z\), 

\(\begin{aligned} z &= - 5y\\ z &= - 5 \times -5\\ \end{aligned}\)

So, 

\(z= 25\)

Another way to find \(x\)is, 

\(\begin{aligned} 3x + w &= 2\\ 3x + 2 &= 2\\ 3x &= 0\\ x &= 0\\ \end{aligned}\)

Conclusion

This article on matrices talks about the basic concepts of matrices, how to properly solve different types of matrices, and the proper manner of presenting the answer.

We also learned about equal matrices, addition, and subtraction of matrices as per the syllabus defined for Secondary 4 Mathematics class. 

This article focuses on both theoretical and practical aspects of learning and is loaded with multiple questions, ranging from easy to hard. Continuous practice and daily revision of concepts can make the journey of learning matrices even simpler.