Overview
Test Series
Matrix operations are basic calculations that we do with matrices, such as adding, subtracting, multiplying, or dividing them. In this topic, we will learn the meaning of these operations, their types, and how they work. We will also look at some important formulas and properties that make solving matrix problems easier. To help you understand better, a few solved examples are included. These examples show how to apply the operations step by step. This guide is especially useful for students preparing for exams, as it explains each concept clearly and helps build a strong foundation in matrix operations.
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A matrix is a way of organizing numbers in a rectangular form using rows (horizontal lines) and columns (vertical lines). These numbers can be real or complex. When we arrange numbers in m rows and n columns, we say it is a matrix of order m × n (read as “m by n”).
A general matrix is written like this:
A = [ a₁₁ a₁₂ ... a₁n
a₂₁ a₂₂ ... a₂n
... ... ... ...
am₁ am₂ ... amn ]
Here, aᵢⱼ means the element in the i-th row and j-th column of the matrix. For example, a₂₃ is the number in the second row and third column.
Another way to write the same matrix is:
A = [aᵢⱼ]ₘₓₙ, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
Matrices are useful tools in many areas of math, science, and computer science to solve equations, represent data, and perform transformations.
We can perform many matrix operations:
The addition of matrices is an operation of adding corresponding elements of two or more two matrices. The matrix addition can be determined only for matrices of the same size (or dimension).
Consider two matrix \(A=[a_{ij}]_{mxn}\text{ and }B=[b_{ij}]_{mxn} \) of order m x n, then the addition of A and B is given by the formula;
\(A + B = [a_{ij}]_{mxn} + [b_{ij}]_{mxn} = [a_{ij} + b_{ij}]_{mxn}\)
The addition of matrices can be done in ways like the element-wise addition of matrices and the direct sum of matrices.
Conditions of Matrix Addition
General Syntax for Addition of Matrices
The general syntax for addition of matrices is as follows:
\(X=\begin{bmatrix}x_{11}&x_{12}\\
x_{21}&x_{22}\end{bmatrix}\)
\(Y=\begin{bmatrix}y_{11}&y_{12}\\
y_{21}&y_{22}\end{bmatrix}\)
\(X+Y=\begin{bmatrix}x_{11}+y_{11}&x_{12}+y_{12}\\
x_{21}+y_{21}&x_{22}+y_{22}\end{bmatrix}\)
There are two types of Addition to the Matrix. Let us learn the two types of methods to obtain matrix sum. The first one is the straightforward method to add the related elements of two or more matrices and another approach is calculating the direct sum of the matrices.
The fundamental properties of matrix addition are equivalent to the addition of real numbers properties. Some of the important properties are; commutative law, associative law, additive inverse, additive identity matrix, etc.
One of the most important conditions for all the above-mentioned properties to hold is that the addition of matrices is defined only if the order of the matrices is identical. We should note that for the addition of matrices, the given matrices need not be square matrices. The addition of rectangular matrices is also carried out if the order of the matrices is identical.
Subtraction between two matrices can be done if they have the same order and the same dimensions. Just like that, they need the same number of columns and rows to get rid of two or more matrices.
Consider two matrix \(A=[a_{ij}]_{mxn}\text{ and }B=[b_{ij}]_{mxn} \) of order m x n, then the addition of A and B is given by the formula;
\(A – B = [a_{ij}]_{mxn} – [b_{ij}]_{mxn} = [a_{ij} – b_{ij}]_{mxn}\)
The addition of matrices can be done in ways like the element-wise addition of matrices and the direct sum of matrices.
Conditions for the subtraction of matrices
General Syntax for subtraction of matrices
The general syntax for subtraction of matrices is as follows:
\(X=\begin{bmatrix}x_{11}&x_{12}\\
x_{21}&x_{22}\end{bmatrix}\)
\(Y=\begin{bmatrix}y_{11}&y_{12}\\
y_{21}&y_{22}\end{bmatrix}\)
\(X-Y=\begin{bmatrix}x_{11}-y_{11}&x_{12}-y_{12}\\
x_{21}-y_{21}&x_{22}-y_{22}\end{bmatrix}\)
The fundamental properties of matrix subtraction are as follows:
Matrix multiplication is a binary operation that produces a matrix from two matrices.
Conditions for Multiplication of Matrix
There are 2 types of Multiplication of Matrix. They are as follows
Scalar Multiplication of Matrix: Upon multiplying a constant to a matrix, it gets multiplied in the complete matrix. Below is an example of matrix multiplication by a constant. The syntax is as follows:
\(x.\ A=x\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}a.\ x&b.\ x\\c.\ x&d.\ x\end{bmatrix}\)
Example: \(2\times{A}=2.\begin{bmatrix}-1&\ \ 2\\\ 4&-3\end{bmatrix}=\begin{bmatrix}-2&\ \ 4\\\ \ 8&-6\end{bmatrix}\)
Matrix Multiplication of Matrix: Whenever we multiply a matrix by another one we are required to obtain the “dot product” of rows of the 1st matrix and columns of the 2nd matrix. The syntax is as follows: \(B=\begin{bmatrix}a&b\\c&d\end{bmatrix}.\begin{bmatrix}e&g\\f&h\end{bmatrix}=\begin{bmatrix}a.\ e+b.f&a.g+b.\ h\\c.\ e+d.f&c.g+d.\ h\end{bmatrix}\)
Example: \(
B=\begin{bmatrix}1&0\\2&4\end{bmatrix}.\begin{bmatrix}2&1\\0&2\end{bmatrix}=\begin{bmatrix}1.\ 2+0.0&1.1+0.\ 2\\2.\ 2+4.0&2.1+4.\ 2\end{bmatrix}=\begin{bmatrix}2&1\\4&10\end{bmatrix}\)
The various properties of matrix multiplication operation in linear algebra in mathematics are as follows.
Properties of Scalar Multiplication of Matrix are as follows:
Any non-singular matrix A = [aij] of order n is said to be invertible or has an inverse if there exists another non-singular square matrix B of order n, such that; \(A ⋅ B = B ⋅ A = I_n\), where I is the identity matrix of order n. Here B is termed as the inverse of A and is denoted by \(A^{-1}\). The inverse of a square matrix says A of order n is presented by: \(A^{-1}=\frac{1}{\left|A\right|}.\ adj\ A\)
Properties of Inverse Matrix are as follows:
For matrices, there is no such thing as division. You can add, subtract, and multiply matrices, but you cannot divide them. There is a related concept, though, which is called “inversion”.
Suppose if there are three matrices such that, A = B.X We need to find the matrix X. Then we multiply both the sides by inverse of B matrix. On the right hand side, the multiplication of B matrix with its inverse cancels out and forms and identity matrix I which after multiplication with matrix X is going to affect it.
Let \(A=\left[a_{i j}\right]\) be a square matrix of order \(n\) and \(C=\left[C_{i j}\right]\) is its co-factored matrix. Then, matrix \(C^{\top}=\left[C_{j i}\right]\) is called the adjoint of matrix \(A\) and is denoted as: \(\operatorname{adj}(A)=C^{\top}=\left[c_{j i}\right], 1 \leq i, j \leq n\)
If A and B are square matrices of order n and in is the corresponding unit matrix then properties of adjoint of a Matrix are:
Let A be a, m × n matrix. Then, the n × m matrix is achieved by interchanging rows and columns of A, this is called transpose of A and is denoted by \(A^{\prime}\) or \(A^{\top}\).
Applications of Matrix Operations refer to the use of matrix addition, subtraction, multiplication, and inversion to solve real-world problems in various fields. These operations help in areas like computer graphics, engineering, data science, and solving systems of equations efficiently.
Application Area |
How Matrix Operations Are Used |
1. Solving Linear Equations |
Matrices help solve systems of equations using methods like Gaussian elimination or inverse matrices. |
2. Computer Graphics |
Used to rotate, scale, and transform images and 3D models using matrix multiplication. |
3. Cryptography |
Matrix operations help encode and decode messages to keep data secure. |
4. Engineering |
Used in structural analysis, circuit design, and robotics to model and calculate real-world systems. |
5. Economics & Business |
Matrices represent and solve problems related to cost, profit, and production planning. |
6. Machine Learning & AI |
Matrices store and process large datasets and perform calculations efficiently. |
7. Physics |
Used to describe transformations, quantum states, and physical systems. |
8. Network Theory |
Matrices represent connectivity in networks (like social networks or internet paths). |
Example 1: Find the matrix B such that A + B = C, where
Matrix A:
A =
[ 2 0 ]
[ 1 4 ]
Matrix C:
C =
[ 3 -1 ]
[ -2 2 ]
We are given:
A + B = C
To find matrix B, we use:
B = C − A
Step-by-step calculation:
B =
[ 3 -1 ]
[ -2 2 ]
−
[ 2 0 ]
[ 1 4 ]
Now subtract corresponding elements:
B =
[ 3 - 2 -1 - 0 ]
[ -2 - 1 2 - 4 ]
B =
[ 1 -1 ]
[ -3 -2 ]
Final Answer:
B =
[ 1 -1 ]
[ -3 -2 ]
Example 2: Add the following matrices: \(\begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix}+\begin{pmatrix} 5 & 0 \\ -6 & 3 \end{pmatrix}\)
A2.
\(
\begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix}+\begin{pmatrix} 5 & 0 \\ -6 & 3 \end{pmatrix}
=\begin{pmatrix} 2+5 & 3+0 \\ 4 – 6 & -1 + 3 \end{pmatrix}
=\begin{pmatrix} 7 & 3 \\ -2 & 2 \end{pmatrix}
\)
Example 3: \(
\text{ If } A = \left[ \begin{array}{rrr} 2 & 1 & 3 \\ -1 & 2 & 0 \end{array} \right] \text{ and } B = \left[ \begin{array}{rrr} 1 & 1 & -1 \\ 2 & 0 & 6 \end{array} \right] \) compute A + B.
A3.
\(\begin{equation*} A + B = \left[ \begin{array}{rrr} 2 + 1 & 1 + 1 & 3 – 1 \\ -1 + 2 & 2 + 0 & 0 + 6 \end{array} \right] = \left[ \begin{array}{rrr} 3 & 2 & 2 \\ 1 & 2 & 6 \end{array} \right] \end{equation*}\)
Example 4:
If A = [ 3 -1 4 ]
[ 2 0 1 ] and
B =[ 1 2 -1 ]
[ 0 3 2 ]
Compute:
(i) 5A
(ii) (1/2)B
(iii) 3A – 2B
(i) 5A =
[15 -5 20]
[10 0 30]
(ii) (1/2)B =
[0.5 1 -0.5]
[0 1.5 1]
(iii) 3A – 2B =
[7 -7 14]
[6 -6 14]
Final Answers:
5A =
[ 15 -5 20 ]
[ 10 0 30 ]
(1/2)B =
[ 0.5 1 -0.5 ]
[ 0 1.5 1 ]
3A – 2B =
[ 7 -7 14 ]
[ 6 -6 14 ]
Example 5: Write down the transpose of each of the following matrices.
\(\begin{equation*} A = \left[ \begin{array}{r} 1 \\ 3 \\ 2 \end{array} \right] \quad B = \left[ \begin{array}{rrr} 5 & 2 & 6 \end{array} \right] \quad C = \left[ \begin{array}{rr} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array} \right] \quad D = \left[ \begin{array}{rrr} 3 & 1 & -1 \\ 1 & 3 & 2 \\ -1 & 2 & 1 \end{array} \right] \end{equation*}\)
A5.
\(\begin{equation*} A^{T} = \left[ \begin{array}{rrr} 1 & 3 & 2 \end{array} \right],\ B^{T} = \left[ \begin{array}{r} 5 \\ 2 \\ 6 \end{array} \right],\ C^{T} = \left[ \begin{array}{rrr} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array} \right], \mbox{ and } D^{T} = D. \end{equation*}\)
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