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.

What are Matrix Operations?

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:

• Addition of Matrix
• Subtraction of Matrix
• Multiplication of Matrix
• Inverse of Matrix
• Division of Matrix
• Adjoint of Matrix
• Transpose of Matrix

Addition of Matrix

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

• Two matrices a and b are added if the order of the given matrices is the same, before adding them.
• Here, by order, we mean the number of the rows and columns should be the same for the matrices.
• Accordingly, we can add the related elements of the matrices.
• But in case if the order is different then the matrix sum is not possible for the given set of matrices.

General Syntax for Addition of Matrices

The general syntax for addition of matrices is as follows:

\(X=\begin{bmatrix}x_{11}&x_{12}\\
x_{21}&x_{22}\end{bmatrix}\)
\(Y=\begin{bmatrix}y_{11}&y_{12}\\
y_{21}&y_{22}\end{bmatrix}\)
\(X+Y=\begin{bmatrix}x_{11}+y_{11}&x_{12}+y_{12}\\
x_{21}+y_{21}&x_{22}+y_{22}\end{bmatrix}\)

Types of Addition of Matrix

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.

• Element-wise Addition of Matrices : In this method for the addition of two matrices, we start adding the elements in individual /first rows and columns to the individual elements in the row and column of the next /second matrix.
  Take two matrices P and Q of the same order i.e. ‘m × n’, where m stands for the number of rows and n for the number of columns of the two matrices, indicated as \(P = [p_{ij}]\) and \(Q = [q_{ij}]\).
  Henceforth the addition of two matrix say P and Q is given as \(P+Q = [p_{ij}] + [q_{ij}] = [p_{ij} + q_{ij}]\).
  Here, ij represents the position of each element in the ith row and jth column. The final dimension of the sum matrix i.e. P + Q is also ‘m × n’.
• Direct Sum – Matrix Addition : In this approach, we calculate the direct sum of two matrices. Here the order of the matrices necessarily need not be the same. Suppose P and Q are two different matrices of orders m × n and a × b, respectively. That is, P has ‘m’ rows and ‘n’ columns and Q has ‘a’ rows and ‘b’ columns. Then, the dimension of the matrix P⊕Q is (m+a) × (n+b). The shows the direct sum of P and Q is expressed as P⊕Q.

Properties of Matrix Addition

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.

• Matrix Commutative Property: If we consider \(P= [p_{ij}]\) and \(Q[q_{ij}]\) as two matrices of the same order, say m x n, then the addition of the two matrices is commutative by the relation P+ Q = Q + P
• Associative Property : If we consider \(P= [p_{ij}]\), \(Q[q_{ij}]\) and \(R[r_{ij}]\) as three matrices of the same order i.e. m x n, then the addition of the three matrices is associative. This can be represented as: P + (Q + R) = (P + Q) + R
• Transpose Property : The transpose of the addition of two matrices is equivalent to the sum of the transposes of the individual matrices. Mathematically written as: \((P+Q)^T=P^T+Q^T\)
• Determinant Property : Similar to the transpose property, the determinant of the addition of two matrices is equivalent to the sum of the determinants of the individual matrices. Mathematically written as: |P + Q| = |P| + |Q|
• Additive identity : If we think of \(P= [p_{ij}]\) as a matrix of order m × n, then the additive identity of P is the zero[O] matrix of order m × n in such a way that P + O = O + P = P. Hence we can say that the zero matrices are the additive identity for the given matrix.
• Additive inverse : If \(X[x_{ij}]\) is any given matrix of order m x n, then the additive inverse of X will be Y (i.e = -X) of the same order. Wherein X + Y = zero(O). Therefore, we can state that the sum of the matrix with its additive inverse is a zero matrix.

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 of Matrix

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

• Two matrices should be of the same order (number of rows=number of columns).
• Subtract the corresponding element of other 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:

• It is a non-commutative operation. If we reverse the order of the matrices and subtract both of them with the same order/dimensions, the result will differ. A-B \(\neq\) B-A
• The negative of matrix A is written as (-A) such that if the addition of matrix with the negative matrix will always produce a null matrix. A+(-A)=0

Multiplication of Matrix

Matrix multiplication is a binary operation that produces a matrix from two matrices.

Conditions for Multiplication of Matrix

• For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
• For matrix products, the matrices should be compatible. This states that two matrices A and B are compatible if the number of columns in A= to the number of rows in B.
• For example, if A is a matrix of order n×m and B is a matrix of order m×p, then one can consider that matrices A and B are compatible.
• Multiplying a matrix of order 4 × 3 by another matrix of order 3 × 4 matrix is valid and it generates a matrix of order 4 × 4.
• Similarly, if we try to multiply a matrix of order 4 × 3 by another matrix 2 × 3. This multiplication of the matrix is not possible as the two matrices do not follow the compatible rule.

Types of 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: \(
B=\begin{bmatrix}1&0\\2&4\end{bmatrix}.\begin{bmatrix}2&1\\0&2\end{bmatrix}=\begin{bmatrix}1.\ 2+0.0&1.1+0.\ 2\\2.\ 2+4.0&2.1+4.\ 2\end{bmatrix}=\begin{bmatrix}2&1\\4&10\end{bmatrix}\)

Properties of Matrix Multiplication

The various properties of matrix multiplication operation in linear algebra in mathematics are as follows.

• Matrix multiplication is generally not commutative – \(AB\ne BA\)
• Matrix multiplication is Associative in Nature – \((AB)C=A(BC)\)
• Matrix multiplication follows the distributive property – \(A(B+C)=AB+AC\) (Distributive law of matrix)
• \(IA=A=AI\), where I is identity matrix for matrix multiplication.
• Product with Scalar \(x(AB)=(xA)B=A(Bx)\) such that x is a scalar
• Transpose of the product of matrices \((AB)^T=B^TA^T\)
• Cancellation law is not applicable \(If AB=AC\ne B=C\)
• Complex Conjugate \((AB)^{*}=B^{*}A^{*}\)
• If AB=0, it does not mean that \(A=0 or B=0\).
• The product of two non zero matrix may be a zero matrix.

Properties of Scalar Multiplication of Matrix

Properties of Scalar Multiplication of Matrix are as follows:

• \(λ(A+B)=λA+λB\)
• \((λ+μ)A=λA+μA\)
• \(λ(μA)=(λμA)=μ(λA)\)
• \((-λA)=-(λA)=λ(-A)\)

Inverse of Matrix

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

Properties of Inverse Matrix are as follows:

• \(\left(A^{-1}\right)^{-1}=A\)
• \(A \cdot B)^{-1}=B^{-1} \cdot A^{-1}\)
• \(\left(A^{k}\right)^{-1}=\left(A^{-1}\right)^{k}, k \in N\)
• \(\operatorname{adj}\left(A^{-1}\right)=(\operatorname{adj}(A))^{-1}\)
• \(\left|A^{-1}\right|=|A|^{-1}\)

Division of Matrix

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.

Adjoint of Matrix

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: \(\operatorname{adj}(A)=C^{\top}=\left[c_{j i}\right], 1 \leq i, j \leq n\)

Properties of Adjoint of Matrix

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:

• \({A} \cdot(\operatorname{adj} {A})=|{A}| \cdot {I}_{{n}}=(\operatorname{adj} {A}) \cdot {A}
  |\operatorname{adj}(A)|=|A|^{n-1}\)
• \( \operatorname{adj}(\operatorname{adj}(A))=|A|^{n-2} \cdot{A},\text{ if } |A| \neq 0\)
• \(\left|adj\left(adj\left(A\right)\right)\right|=\left|A\right|^{\left(n-1\right)^2}\)
• \(\operatorname{adj}\left(A^{\top}\right)=(\operatorname{adj} A)^{\top}\)
• \(\operatorname{adj}(A \cdot B)=(\operatorname{adj} B) \cdot(\operatorname{adj} A)\)
• \(\operatorname{adj}\left(A^{m}\right)=(\operatorname{adj} A)^{m}, m \in N\)
• \(\operatorname{adj}(k \cdot A)=k^{n-1} \cdot(\operatorname{adj} A), k \in R\)

Transpose of Matrix

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 \(A^{\top}\).

Properties of Transpose of Matrix

• If A is a matrix of order m × n, then (A’)’ = A
• If A and B are matrices of the same order m × n, then (A ± B)’ = A’ ± B’.
• If k ∈ R is a scalar and A is a matrix of order m × n, then (k × A)’ = k × A’
• If A and B are matrices of order m × n and n × p respectively, then (AB)’ = B’ × A’.

Important Points of Matrix Operations

• We use matrices to list data or to represent systems. Because the entries are numbers, we can perform Matrix Operations. We add or subtract matrices by adding or subtracting corresponding entries.
• We can add, subtract, multiply, and even, in a sense, “divide” matrices to organize our work.

Applications of Matrix Operations

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.

Solved Examples on Matrix Operations

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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