chevron icon chevron icon chevron icon chevron icon

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 \(-3\).

 

\(8 = 4m\)

\(\frac {1}{2}n = -3\)

\(4m = 8\)

\(n=-3 \div \frac {1}{2}\)

\(m = 2 \)

\(n  = -6\)

 

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.

Continue Learning
Sets: Venn Diagrams, Intersections & Union Probability of Combined Events
Statistical Data Analysis Matrices
Vectors
Resources - Academic Topics
icon expand icon collapse Primary
icon expand icon collapse Secondary
icon expand icon collapse
Book a free product demo
Suitable for primary & secondary
select dropdown icon
Our Education Consultants will get in touch with you to offer your child a complimentary Strength Analysis.
Book a free product demo
Suitable for primary & secondary
Claim your free demo today!
Claim your free demo today!
Arrow Down Arrow Down
Arrow Down Arrow Down
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
Geniebook CTA Illustration Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Geniebook CTA Illustration Geniebook CTA Illustration
Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Get a free diagnostic report of your child’s strengths & weaknesses!
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
Gain access to 300,000 questions aligned to MOE syllabus
Trusted by over 220,000 students.
Trusted by over 220,000 students.
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
media logo
Geniebook CTA Illustration
Geniebook CTA Illustration
Geniebook CTA Illustration
Geniebook CTA Illustration Geniebook CTA Illustration
icon close
Default Wrong Input
Get instant access to
our educational content
Start practising and learning.
No Error
arrow down arrow down
No Error
*By submitting your phone number, we have
your permission to contact you regarding
Geniebook. See our Privacy Policy.
Success
Let’s get learning!
Download our educational
resources now.
icon close
Error
Error
Oops! Something went wrong.
Let’s refresh the page!