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108 LearnersLast updated on November 25, 2025
Matrix Operations
Matrix operations are a set of mathematical rules or procedures that can be performed on matrices. They help combine two or more matrices and are essential in fields like physics, algebra, and computer graphics. In this article, we will learn different matrix operations, their properties, and real-life applications.
What are the Matrix Operations?
Matrix operations are mathematical processes that help perform calculations on one or more matrices, often resulting in a new matrix. Some basic algebraic operations are used in matrix operations, such as addition, subtraction, and multiplication. Matrix operations can also change a matrix using its transpose or inverse to transform its properties or structure.
Properties of Matrix Addition
Matrix addition involves adding corresponding elements of matrices.
An important rule in matrix addition is that if you are adding two or more matrices, they all must be of the same order.
It means all matrices should have the same number of rows and columns.
While adding two matrices A and B, each element in matrix A should be added to the corresponding element in matrix B.
Matrix addition has several key properties, including:
- Commutative
- Associative
- Identity of Matrix
- Additive Inverse
Commutative Property
The commutative property states that if two matrices A and B are of the same size, then: A + B = B + A. It means when two matrices of the same size are added, the order of the matrices doesn’t matter as the result will be the same.
Associative Property
This property is used when adding three matrices; the matrices can be grouped in any way, but the three matrices must be in the same dimensions, and this does not affect the result.
(A + B) + C = A (B + C)
Identity of Matrix
The identity of a matrix is also known as the zero matrix. When a matrix A is added to the zero matrix 0 (where all elements are zero), the result is the original matrix: A+0=A.
Additive Inverse
The additive inverse of a matrix A is a matrix −A, where each element is the negative of the corresponding element in A, such that A +(−A)=0.
What is Subtraction of Matrices?
Matrix subtraction involves subtracting corresponding elements of two matrices. A key requirement for matrix subtraction is that the matrices must have the same dimensions, i.e., the same number of rows and columns.
What is Scalar Multiplication of Matrices?
Scalar multiplication of a matrix involves multiplying all elements in the matrix by a scalar. Scalar multiplication can be performed on any type of matrix, whether it is rectangular or square. It is used for transformation and solving matrix equations.
Properties of Matrix Multiplication
Matrix multiplication means you're combining two matrices in a specific way. Scalar multiplication is when every element of a matrix is multiplied by the same number, which is called a scalar.
So if you have a number k (the scalar), and a matrix A, then:
kA=a new matrix where each number in A is multiplied by k
Let’s say:
- A and B are two matrices of the same size
- k and l are scalar numbers (just regular constants)
1. Distributive Property Over Matrix Addition
You can either add the matrices first, then multiply the result by k,
Or multiply each matrix by k first, then add the results — it gives the same answer!
k(A+B)=kA+kB
2. Distributive Property Over Scalar Addition
If you add the scalars first and then multiply, it’s the same as splitting the scalar — multiply AA by each scalar separately, then add.
(k+l)A=kA+lA
3. Associative Property of Scalar Multiplication
No matter how you group or order the scalars, you’ll get the same result. You can multiply the scalars first, or apply them one at a time to the matrix — it all works out the same.
(kl)A=k(lA)=l(kA)
Transpose Operation of a Matrix
The transpose of a matrix is obtained by swapping its rows with columns. The symbol of the transpose operation of a matrix is At or AT.
Inverse Operation of a Matrix
The inverse of a matrix, written as A⁻¹, is a matrix that reverses the effect of matrix A.
For square matrices, the inverse only exists if the determinant is not zero. For a 2 × 2 matrix, we can find the inverse using a specific formula. When you multiply A by its inverse—from either side—you always get the identity matrix (I), which acts like the number 1 in matrix multiplication.
So, A × A⁻¹ = A⁻¹ × A = I.
Common Mistakes and How to Avoid Them in Matrix Operations
Matrix operations can be confusing when students are learning them for the first time. Many students make small mistakes, like adding the wrong-sized matrices or mixing up multiplication rules. Here for some examples given below:
Mistake 1
Adding or Subtracting Matrices of Different Sizes
Students might make the mistake of adding or subtracting matrices of different sizes. Before adding or subtracting, always double-check that both matrices have the same size. If they don’t match, the operation isn’t possible.
Mistake 2
Multiplying Matrices Incorrectly
Sometimes students might simply multiply each matching element of two matrices. This is not how two matrices are multiplied. Even before multiplying, we should make sure the number of columns in the first matrix matches the number of rows in the second. This condition has to be satisfied when two matrices are multiplied because we need to perform row-by-column multiplication. For example, when multiplying two matrices A and B:
The wrong calculation is:
This is wrong because here element-to-element multiplication has been performed.
The right calculation is: 
Mistake 3
Mixing Up Matrix Multiplication with Scalar Multiplication
Some students confuse multiplying a matrix by another matrix with multiplying a matrix by a single number (a scalar). If you're multiplying a matrix by a number, just multiply all the elements individually with the scalar. Let’s look at an example for better understanding. Let’s multiply a matrix A with a scalar k, where:
and k =3. The correct scalar multiplication is:
Mistake 4
Incorrect Transpose
Students often make mistakes while taking the transpose of a matrix. They either change the positions of the elements randomly or forget how to flip rows and columns. When you transpose a matrix, just remember this rule: the element that was in row A, column B moves to row B, column A. In simple terms, rows become columns and columns become rows—neatly and in order.
Mistake 5
Ignoring the Order in Matrix Multiplication
Students often assume that matrix multiplication is commutative, i.e., A × B is the same as B × A. This is not true for matrices, as the order matters in matrix multiplication. Always pay attention to the order and the dimensions of the matrices.
Real-Life Applications of Matrix Operations
Matrix operations have many real-life applications in various fields. In fact, this mathematical concept plays an important role in our everyday lives. Here are some real-life applications of matrix operations:
Computer Graphics and Video Games: Rotation is used to turn 3D objects left, right, or rotate them around. Translation moves them from one place to another. Scaling makes them look bigger or smaller.
Medical Image: Medical images like CT scans, MRI, and ultrasound data are obtained using matrix operations. These images are stored as grids of pixels, in other words, matrices of numbers.
Sensor Fusion: Matrices help in sensor fusion to collect the data and combine it from multiple sensors for accurate measurement.
Communication Systems: Signal Processing, audio, and radio signals are filtered using matrix-based algorithms like the Fourier transform.
Robotics: Matrices perform spatial transformations in robotics, allowing robots to determine their position and orientation.
Solved Examples on Matrix Operations
Problem 1
A = 34 76 B = 21 58 Find A + B
\(A + B = \begin{bmatrix} 5 & 5 \\ 12 & 14 \end{bmatrix}\)
Explanation
Add corresponding elements like the first element in the first row of matrix A is added to the first element in the first row of matrix B, and so on.
\(A + B = \begin{bmatrix} 4+1 & 6+8 \\ 3+2 & 7+5 \end{bmatrix}
= \begin{bmatrix} 5 & 14 \\ 5 & 12 \end{bmatrix}\)
Problem 2
A = 69 34 B = 25 31 Find = A - B
\(A - B = \begin{bmatrix} 4 & 4 \\ 0 & 3 \end{bmatrix}\)
Explanation
Subtract the corresponding elements of matrices A and B
\(A - B = \begin{bmatrix} 9-5 & 4-1 \\ 6-2 & 3-3 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ 4 & 0 \end{bmatrix}\)
Problem 3
A = 31 42 B = 12 20
A × B = \(\begin{bmatrix} 4 & 4 \\ 10 & 8 \end{bmatrix}\)
Explanation
Multiply each element in the row of A with the corresponding element in the column of B, then add them. Repeat the process for all the elements.
\(A \times B = \begin{bmatrix} 1 \times 2 + 2 \times 1 & 1 \times 0 + 2 \times 2 \\ 3 \times 2 + 4 \times 1 & 3 \times 0 + 4 \times 2 \end{bmatrix}\)
A × B = \(\begin{bmatrix} 4 & 4 \\ 10 & 8 \end{bmatrix}\)
Problem 4
Find At (Transpose of A), A = 21 43
\(A^T = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)
Explanation
Swap rows and columns
\(A^T = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)
Problem 5
Find: k × A, A = 42 0-1 and k = 3
\(3A = 3 \times \begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 12 & 0 \end{bmatrix}\)
Explanation
Multiply every element by 3:
3A = \(\begin{bmatrix} 6 & -3 \\ 12 & 0 \end{bmatrix}\)
FAQs
1.What is a matrix operation?
2.When can I add or subtract two matrices?
3.How do you multiply two matrices?
4.Can every matrix be inverted?
5.Why do we use matrix operations in real life?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
