Matrices Class 12 Notes

Matrix is one of the important concepts of Mathematics and one of the most powerful tools, which has various applications such as in solving linear equations, budgeting, sales projection, cost estimation, etc. Matrices for class 12 covers the important concepts in matrices, such as types, order, matrix elementary transformation operations and so on. Students can get a detailed explanation of matrix concepts here. Matrices for class 12 helps the students with their higher studies, as it covers all the basic topics. Go through the notes on class 12 matrices to score good marks in the examinations.

Matrices for Class 12 Topics

The topics covered in matrices for class 12 include the following topics:

• Introduction
• Matrix
• Types of Matrices  
• Operations on Matrices 
• Transpose of a Matrix 
• Symmetric and Skew Symmetric Matrices
• Elementary Operation (Transformation) of a Matrix 
• Invertible Matrices

Matrices Definition

A matrix is a function that consists of an ordered rectangular array of numbers. The numbers in the array are called the entities or the elements of the matrix. The horizontal array of elements in the matrix is called rows, and the vertical array of elements are called the columns. If a matrix has m rows and n columns, then it is known as the matrix of order m x n.

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To know, how to determine the order of a matrix, visit here.

Types of Matrices

Depending upon the order and elements, matrices are classified as:

• Column matrix
• Row matrix
• Square matrix
• Diagonal matrix
• Scalar matrix
• Identity matrix
• Zero matrix

Let’s understand the definition of all these types of matrices along with examples here.

Equality of Matrices

Two matrices are said to be equal if-

(i) The order of both the matrices are the same

(ii) Each element of one matrix is equal to the corresponding element of the other matrix

Operations on Matrices

In Chapter 3 of Class 12 Matrices, certain operations on matrices are discussed, namely, the addition of matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices.

Also, Check:

• Addition of Matrices
• Subtraction of matrices
• Matrix multiplication

Transpose of a Matrix

 If A = [a_ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A′ or (A^T ). 

In other words, if A = [a_ij] m × n , then A′ = [a_ji] n × m . 

Example:

Matrix A = \(\begin{bmatrix} 2 &1&3\\ -4&0&5\\ \end{bmatrix}\)

Transpose of A = A^T = \(\begin{bmatrix} 2 &-4\\ 1& 0\\ 3&5\\ \end{bmatrix}\)

To learn more about transpose of a matrix, visit here.

Symmetric and Skew Symmetric Matrices

 A square matrix A = [a_ij] is said to be symmetric if the transpose of A is equal to A, that is, [a_ij] = [a_ji] for all possible values of i and j.

A square matrix A = [a_ij] is a skew-symmetric matrix if A′ = – A, that is a_ji = – a_ij for all possible values of i and j. Also, if we substitute i = j, we have a_ii = – a_ii and thus, 2a_ii = 0 or a_ii = 0 for all i’s. Therefore, all the diagonal elements of a skew symmetric matrix are zero.

To understand the symmetric and skew-symmetric matrix in detail, visit here.

Elementary Operation (Transformation) of a Matrix 

There are six operations (transformations) on a matrix, three of which are due to rows, and three are due to columns, known as elementary operations or transformations.

1. The interchange of any two rows or two columns.
2. The multiplication of the elements of any row or column by a non zero number.
3. The addition to the elements of any row or column, the corresponding elements of any other row or column are multiplied by any non zero number.

Learn more about the elementary operations of the matrix here.

Invertible Matrices 

Suppose a square matrix A of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A, and it is denoted by A^-1. Also, matrix A is said to be an invertible matrix here.

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Matrices for Class 12 Examples

Example 1: 

If \(\begin{bmatrix} x+3 & z+4 & 2y-7\\ -6 & a-1 & 0\\ b-3 & -21 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 6 & 3y-2 \\ -6 & -3 & 2c +2\\ 2b+4 & -21 & 0 \end{bmatrix}\), then find the value of a, b, c, x, y, and z.

Solution:

It is given that, the two matrices are equal. Therefore, the corresponding elements present in matrices should be equal to each other. By comparing the corresponding elements in the matrices, we get:

x+3 = 0.

⇒ x = -3

z +4 = 6

⇒ z = 6-4

⇒ z = 2

2y-7 = 3y-2

⇒3y-2y =-7+2

 ⇒y = -5

a-1 = -3

⇒a = -3+1

 ⇒a=-2

2c+2 = 0

⇒2c = -2

⇒ c = -1

b-3 = 2b+4

⇒2b-b = -3-4

⇒ b = -7

Therefore, the values of the variables are:

a = -2

b = -7

c = -1

x = -3

y = -5

z = 2

Example 2:

If \(A=\begin{bmatrix} 1\\ 3\\ -6\\ \end{bmatrix}\) and \(B=\begin{bmatrix} -2&4&5 \end{bmatrix}\), then verify that (AB)^T = B^TA^T.

Solution:

Given,

\(A=\begin{bmatrix} 1\\ 3\\ -6\\ \end{bmatrix}\) and \(B=\begin{bmatrix} -2&4&5 \end{bmatrix}\)

\(AB=\begin{bmatrix} 1\\3\\-6 \end{bmatrix}\begin{bmatrix} -2&4&5 \end{bmatrix}\\=\begin{bmatrix} -2&4&5\\ -6&12&15\\ 12&-24&-30 \end{bmatrix}\)

Now, we need to calculate the transpose of AB.

\((AB)^T=\begin{bmatrix} -2&-6&12\\ 4&12&-24\\ 5&15&-30 \end{bmatrix}\)

\(A^T=\begin{bmatrix} 1&3&-6 \end{bmatrix}\)

And

\(B^T=\begin{bmatrix} -2\\ 4\\5 \end{bmatrix}\)

\(B^TA^T=\begin{bmatrix} -2\\ 4\\5 \end{bmatrix}\begin{bmatrix} 1&3&-6 \end{bmatrix}=\begin{bmatrix} -2&-6&12\\ 4&12&-24\\ 5&15&-30 \end{bmatrix}\)

Therefore, (AB)^T = B^TA^T.

Hence verified.

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