Matrices Class 12 NCERT Solutions defines matrices as arrays containing elements or entries used to represent coefficients of linear equations. They have rows that are represented as ‘i’ and columns that are represented as ‘j’. Thus, a matrix is of ‘i x j’ order where ‘i’ is the number of rows and ‘j’ is the number of columns.

For example, consider the following linear equations:

3x + 6y + z = 4

4x + 7y + 2z = 2

5x + 8y + 9z = 5

Here, the matrix of the LHs is:

While the matrix of the RHS is:

Solved examples for Class 12 Maths NCERT Solutions, Chapter 3 Matrices:

Example 1: Find the values of x, y, and z in the following matrix equation.

LHS:

RHS:

Answer: 

By definition of matrix, we get the following linear equations from the matrices above:

x + y + a = 9 (1)

x + a = 5 (2)

y + a = 7 (3)

Adding equations (2) and (3), we get,

x + y + 2a = 12 (4)

Subtracting (1) from (4), we get a = 3.

From (2) we get x = 2 and from (3) we get y = 4

Hence, x = 2, y = 4, and a =3

• Row matrix 
• A matrix having only one row.

For example:

[ 2 4 5 ]

The above matrix is of the order 1 x 3.

• Column matrix
• A matrix having only one column.

For example:

The above matrix is of the order 4 x 1.

• Square matrix
• ‘i’ represents the rows while ‘j’ represents columns.
• In this matrix, i = j, i.e., the number of columns is equal to the number of rows.

For example:

The above matrix is of the order 4 x 4.

• Zero matrix
• A matrix whose elements are zeros. It is also known as a null matrix.
• All other elements are zeroes.

For example:

[ 0 0 0 ]

The above matrix is of the order 1 x 3.

• Diagonal matrix
• A matrix with non-zero values along the diagonal.

For example:

The above matrix is of the order 3 x 3.

• Identity matrix
• A matrix with one as the diagonal element.
• All other elements are zero.

For example:

The above matrix is of the order 3 x 3.

• Scalar matrix
• It is a matrix in which all the diagonal elements are equal.

For example:

The above matrix is of the order 4 x 4.

• Triangular matrix
• Upper triangular matrix: A matrix with non-zero elements in its upper right or left half.

For example:

• Lower triangular matrix: A matrix with non-zero elements in its lower right or left half. For example:

Addition of matrices

• Here, each element is added individually.
• Example: [ 2 4 6 ] + [ 1 3 4] = [ 3 7 10]

Subtraction of matrices

• Here, each element is subtracted individually.
• Example: [ 2 4 6 ] - [ 1 3 4] = [ 1 1 2]

Negative of a matrix

• Here, each element is assigned opposite signs.
• Example: The negative of the matrix A = [ -1 2 3 ] is -A = [ 1 -2 -3 ]

Multiplication of a matrix by a scalar

• Here, each element is multiplied individually.
• Example: [ 1 2 3 ] x [ 4 5 6 ] = [ 4 10 18 ]
• Properties of multiplication of a matrix by a scalar (number) are
• (k + l) x A = kA + lA
• K (A + B) = kA + kB
• (kl) x A = k(lA) = l(kA)
• 1A = A
• (-1)A = -A

Operations between matrices of completely different orders can also be performed provided the corresponding elements in each of the matrices are operated with their respective element in the other matrix.

It is obtained by interchanging the rows and columns.

For example, the transpose of the matrix A = [ 5 2 9 8 ] is AI=

The following are the properties of transposes:

o  (A’)’ = A

o  (kA)’ = k(A)’

o  (-A)’ = -A’

o  (A - B)’ = A’ - B’

o  (A + B)’ = A’ + B’

(AB)’ = B’ A’

Symmetric matrix

They are matrices with similar values even upon transposition.

For example,

Skew-symmetric matrix

These matrices have negative elements upon transposition.

For example,

In the above image, the following can be deduced:

• B can be obtained from A by interchanging the first and second rows.
• C can be obtained from A by interchanging the first and third columns.
• D can be obtained from A by multiplying each element of the first row by 3.
• E can be obtained from A by multiplying each element of the second column by 5.
• F can be obtained from A by adding to the elements of the second row, 4 times the corresponding elements of the first row.
• G can be obtained from A by adding to the elements of the third column, (-3) times the corresponding elements of the first column.
• B, D, and F can be obtained from A by applying elementary row operations whereas C, E, and G can be obtained from A by applying elementary column operations.

Elementary operation of a matrix can be any one of the following three types:

• The interchanging of two columns or rows.
• The multiplication of the elements in a row or column by any real number except zero.
• The multiplication of the elements in a row or column by any real number except zero, the result of which is added to another row or column.

Two matrices are said to be invertible if their product is an identity matrix.

For example:

Solved examples for Matrices Class 12 NCERT Solutions:

• Multiply the following matrix:
• [ 3 4 5 ] x 3[ 2 4 5 ] = [18 48 75 ]
• 2 [ 1 2 3 ] x [ 2 3 4 ] x [ 3 4 5 ] = [ 12 48 120 ]

• What is the invertible matrix of the following matrix:

The invertible matrix can be calculated as follows:

Maths Chapter 3 Matrix is an essential subject for class 12 as well as a difficult one with multiple problems, diagrams and concepts. 

Key points of Matrices Class 12 Ncert Solutions:

• Easy language and eye-catching formats.
• Topics as per the latest syllabus pattern that aid students in revising the notes in minimum time with maximum accuracy.
• NCERT Class 12 Maths Chapter 3 Matrix notes are as per the guidelines of the CBSE syllabus.
• After studying NCERT Class 12 Maths Chapter 3 Matrix notes, students will be more confident when presented with other enormous books.
• The NCERT Class 12 Maths Chapter 3 Matrix notes cover all necessary formulas and concepts presented in the chapter.
• These notes will evidently save time when students prepare for the exams.

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