NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.4 Determinants

Class 12 Maths Chapter 4 Exercise 4.4 Determinants:- Chapter 4 of Class 12 Maths, titled "Determinants," covers the basic concepts and applications of determinants, which are crucial in various areas of mathematics. Exercise 4.4 focuses on calculating determinants, especially for 3x3 matrices, and understanding their properties. It includes methods for finding determinants, the significance of determinants in solving linear equations, and their role in determining the invertibility of matrices.  Go through the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.4 Determinants below.

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NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.4 Determinants

Here are the Class 12 Maths Chapter 4 Exercise 4.4 Determinants NCERT Solutions.

Question 1. Write minors and cofactors of the elements of the following determinants:

(i)

(ii)

Solution :
(i) Let

Minor of element aij is Mij.

∴M11 = minor of element a11 = 3

M12 = minor of element a12 = 0

M21 = minor of element a21 = −4

M22 = minor of element a22 = 2

Cofactor of aij is Aij = (−1)i + j Mij.

∴A11 = (−1)1+1 M11 = (−1)2 (3) = 3

A12 = (−1)1+2 M12 = (−1)3 (0) = 0

A21 = (−1)2+1 M21 = (−1)3 (−4) = 4

A22 = (−1)2+2 M22 = (−1)4 (2) = 2

(ii) Let

Minor of element aij is Mij.

∴M11 = minor of element a11 = d

M12 = minor of element a12 = b

M21 = minor of element a21 = c

M22 = minor of element a22 = a

Cofactor of aij is Aij = (−1)i + j Mij.

∴A11 = (−1)1+1 M11 = (−1)2 (d) = d

A12 = (−1)1+2 M12 = (−1)3 (b) = −b

A21 = (−1)2+1 M21 = (−1)3 (c) = −c

A22 = (−1)2+2 M22 = (−1)4 (a) = a

Check out, NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 Determinants

Question 2. Write minors and cofactors of the elements of the following determinants:

 

Solution :

A11 = cofactor of a11= (−1)1+1 M11 = 1

A12 = cofactor of a12 = (−1)1+2 M12 = 0

A13 = cofactor of a13 = (−1)1+3 M13 = 0

A21 = cofactor of a21 = (−1)2+1 M21 = 0

A22 = cofactor of a22 = (−1)2+2 M22 = 1

A23 = cofactor of a23 = (−1)2+3 M23 = 0

A31 = cofactor of a31 = (−1)3+1 M31 = 0

A32 = cofactor of a32 = (−1)3+2 M32 = 0

A33 = cofactor of a33 = (−1)3+3 M33 = 1

A11 = cofactor of a11= (−1)1+1 M11 = 11

A12 = cofactor of a12 = (−1)1+2 M12 = −6

A13 = cofactor of a13 = (−1)1+3 M13 = 3

A21 = cofactor of a21 = (−1)2+1 M21 = 4

A22 = cofactor of a22 = (−1)2+2 M22 = 2

A23 = cofactor of a23 = (−1)2+3 M23 = −1

A31 = cofactor of a31 = (−1)3+1 M31 = −20

A32 = cofactor of a32 = (−1)3+2 M32 = 13

A33 = cofactor of a33 = (−1)3+3 M33 = 5

Check out, NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5 Determinants

Question 3. Using cofactors of elements of second row, evaluate:

Solution :

We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.

∴Δ = a21A21 + a22A22 + a23A23 = 2(7) + 0(7) + 1(−7) = 14 − 7 = 7

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Question 4. Using cofactors of elements of third column, evaluate:

Solution :

Question 5. If and Aij is Cofactors of aij, then value of Δ is given by

Solution :

We know that:

Δ = Sum of the product of the elements of a column (or a row) with their corresponding cofactors

∴Δ = a11A11 + a21A21 + a31A31

Hence, the value of Δ is given by the expression given in alternative D.
Option (D) is correct.

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Class 12 Maths Chapter 4 Exercise 4.4 Determinants Summary

Minors and Cofactors

• Minor (Mij): This is the determinant of the sub-matrix obtained by excluding the i-th row and j-th column of the original matrix.

• Cofactor (Aij): The minor (Mij) of element aij is multiplied by (-1)^(i+j). In simpler terms, the cofactor is the minor with a sign determined by the sum of the row and column indices (i + j).

Expanding Determinants

• Exercise 4.4 deals with expanding determinants of any order using minors and cofactors. The determinant of an n x n matrix can be expanded along any row or column. The expansion along a specific row (let's say the i-th row) is given by:

Determinant = Σ(Aij * Mij) (where j goes from 1 to n)

Here, Aij is the cofactor of element aij in the i-th row, and Mij is the minor of aij. This basically means you take each element in the chosen row, multiply it by its corresponding cofactor, and add all those products together.

Class 12 Maths Chapter 4 Exercise 4.4 Determinants FAQs

Q1. How is the determinant of a 2x2 matrix calculated?

Ans. For a 2x2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}(ac​bd​), the determinant is calculated as ad−bcad - bcad−bc.

Q2. What is the significance of the determinant of a matrix?

Ans. The determinant provides information about the matrix, such as whether it is invertible. A matrix is invertible if and only if its determinant is non-zero.

Q3. How do you calculate the determinant of a 3x3 matrix?

Ans. For a 3x3 matrix (abcdefghi)\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}​adg​beh​cfi​​, the determinant is calculated as a(ei−fh)−b(di−fg)+c(dh−eg)a(ei - fh) - b(di - fg) + c(dh - eg)a(ei−fh)−b(di−fg)+c(dh−eg).

Q4. What is the cofactor of an element in a matrix?

Ans. The cofactor of an element in a matrix is the signed minor of that element. It is calculated by taking the determinant of the submatrix formed by deleting the row and column of the element, and then multiplying by (−1)i+j(-1)^{i+j}(−1)i+j where iii and jjj are the row and column indices of the element.

Q5. How do you find the determinant of a larger matrix, such as 4x4 or bigger?

Ans. The determinant of a larger matrix is found using the method of cofactor expansion, which involves breaking down the matrix into smaller matrices and calculating their determinants recursively.