A Matrix is an organized rectangular array comprising numbers, symbols, or expressions that are systematically arranged in rows and columns. It is typically denoted by uppercase letters and is used in Mathematics and Computer science to represent data or perform operations such as addition, subtraction, multiplication, and more.
A matrix is termed a Column Matrix if it contains only one column.
Example:
A matrix is considered a Row Matrix if it consists of only one row.
Example - A = [1 5 7]
A square matrix is one where the number of rows equals the number of columns.
Example :
A square matrix B = [bij] of order m × m is classified as a Diagonal Matrix if all its non-diagonal elements are zero. In other words, a matrix B = [bij] of size m × m is designated as a diagonal matrix if bij = 0 whenever i ≠ j.
A Scalar Matrix is a square matrix where all diagonal elements are equal (scalar), and all non-diagonal elements are zero.
A square matrix in which ones are on the main diagonal and zeros anywhere else is known as an identity matrix. It is denoted as In for an n × n matrix.
A zero matrix, denoted as 0, is a matrix in which all elements are zero.
An orthogonal matrix is a square matrix where the transpose is equal to its inverse.
AT = A–1
To add two matrices, you add corresponding elements in the same position in each matrix. Here is the general formula for adding two matrices:
If two matrices A = [aij] and B=[bij] have same order m×n, then their sum A + B is given by:
A + B=[aij] + [bij] = [aij +bij]
Example:
Multiplying a matrix by a scalar involves multiplying each element of the matrix by the scalar value. For example, if we have a matrix A and a scalar k, the product kA is obtained by multiplying each element of A by k.
Example:
Matrix multiplication is a binary operation that combines two matrices to produce a new matrix. For matrix multiplication to be defined the number of columns in matrix A should match the number of rows in matrix B.
Let's consider two matrices, A and B:
To perform matrix multiplication between matrices A and B, we compute each element of the resulting matrix C by taking the scalar product of the corresponding row of matrix A and the respective column of matrix B:
To multiply A and B, we follow the rule that the (i,j)th element of the resulting matrix C is obtained by taking the scalar product of the ith row of A and the jth column of B:
where
c11 = a11⋅b11 + a12⋅b21,
c12 = a11⋅b12 + a12⋅b22,
c13 = a11⋅b13 + a12⋅b23,
c21 = a21⋅b11 + a22⋅b21,
c22 = a21⋅b12 + a22⋅b22,
c23 = a21⋅b13 + a22⋅b23,
c31 = a31⋅b11 + a32⋅b21,
c32 = a31⋅b12 + a32⋅b22,
c33 = a31⋅b13 + a32⋅b23.
The transpose of a matrix is generated by interchanging its rows and columns matrix.. For a matrix A with dimensions m×n, the transpose AT has dimensions n×m.
Let's consider a matrix A with elements aij:
The transpose AT is formed by interchanging rows and columns:
In other words, the (i, j)th element of AT is the (j, i)th element of A, i.e.,
( AT)ij= Aji.
The transpose operation is important in various matrix operations and has several properties, such as:
A square matrix A = [aij] is said to be, symmetric if, aij =aij ∀ i and j(conjugate elements are equal).
Hence for symmetric matrix A = A T.
Square Matrix A = [aij] is said to be skew symmetric if, aij = –aij ∀ i and j(the pair of conjugate elements are additive inverse of each other). For a skew symmetric matrix, A = – AT.
A square matrix is A = [aij] is said to be invertible if |A| ≠ 0, and AA–1 = A–1A = I.
Example 1: If & and A+ B –D = 0 (Zero matrix), then D matrix will be ?
(A)
(B)
(C)
(D)
Ans. (C)
Solution:
∴
⇒ –a = 0 ⇒ a = 0, 1 –b = 0 ⇒ b = 1,
⇒ 3–c = 0 ⇒ c = 3,7 – d = 0 ⇒ d = 7,
⇒ 5 – e = 0 ⇒ e = 5,6 – f = 0 ⇒ f = 6
Example 2: If A,B are two matrices such that A+B=, A-B=, then find AB.
Solution:
Given A+B= …(i) &
…(ii) &
Adding (i) & (ii)
Subtracting (ii) from (i)
Now A B=
Example 3: If A=, show that , where k is any positive integer.
Solution: We have
and
Thus, it is true for indices 2 and 3. Now assume
Then,
Example 4: If A is symmetric as well as skew symmetric matrix, then A is-
(A) diagonal matrix (B) null matrix
(C) triangular matrix (D) none of these
Ans. (B)
Solution: Let A = [aij] Since A is skew symmetric aij = –aji
for i = j, aii = –aii ⇒ aii =0
for i ≠ j, aij = – aji [∵A is skew symmetric ] and aij = aji [∵ A is symmetric]
∴ aij = 0 for all i ≠ j
So, aij = 0 for all ‘i’ and ‘j’ i.e. A is a null matrix.
Example 5: If A=, then adj (adj A) is equal to
(A)
(B)
(C)
(D) None of these
Ans. (B)
Solution:
|A|=
Now adj (adj A) = |A|3–2A
=
Q1. What is an invertible matrix?
Ans: An invertible matrix, also known as a nonsingular matrix, is a square matrix that has an inverse. An invertible matrix A satisfies the condition that there exists another matrix A−1 such that AA−1 =A−1A = I, where I represent the identity matrix.
Q2. How do you find the determinant of a matrix?
Ans: The determinant of a square matrix can be found using various methods such as cofactor expansion.
For a 2 × 2 matrix , the determinant is given by ad−bc. For larger matrices, the process involves recursive calculations based on cofactors.
(Session 2025 - 26)