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NCERT Solutions Exercise 4.2: Determinants | Mathematics (Maths) Class 12 - JEE PDF Download

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Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
Page 2


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
Page 3


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
By applying C3 ? C3 + C2, we have: 
Here, two columns C1 and C3 are proportional. 
? = 0. 
Question 5:  
Using the property of determinants and without expanding, prove that: 
Answer 
Applying R2 ? R2 - R3, we have: 
Page 4


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
By applying C3 ? C3 + C2, we have: 
Here, two columns C1 and C3 are proportional. 
? = 0. 
Question 5:  
Using the property of determinants and without expanding, prove that: 
Answer 
Applying R2 ? R2 - R3, we have: 
Applying R1 ?R3 and R2 ?R3, we have: 
Applying R1 ? R1 - R3, we have: 
Applying R1 ?R2 and R2 ?R3, we have: 
From (1), (2), and (3), we have: 
Page 5


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
By applying C3 ? C3 + C2, we have: 
Here, two columns C1 and C3 are proportional. 
? = 0. 
Question 5:  
Using the property of determinants and without expanding, prove that: 
Answer 
Applying R2 ? R2 - R3, we have: 
Applying R1 ?R3 and R2 ?R3, we have: 
Applying R1 ? R1 - R3, we have: 
Applying R1 ?R2 and R2 ?R3, we have: 
From (1), (2), and (3), we have: 
Hence, the given result is proved. 
Question 6:  
By using properties of determinants, show that: 
Answer 
We have, 
Here, the two rows R1 and R3 are identical. 
? = 0. 
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FAQs on NCERT Solutions Exercise 4.2: Determinants - Mathematics (Maths) Class 12 - JEE

1. What is a determinant?
Ans. A determinant is a scalar value that can be calculated from a square matrix. It is used in various mathematical calculations, including solving systems of equations, finding inverses of matrices, and determining whether a matrix is invertible or singular. The determinant is calculated by summing the products of the elements of a matrix in a specific pattern.
2. How do you calculate the determinant of a 2x2 matrix?
Ans. To calculate the determinant of a 2x2 matrix, you need to subtract the product of the top left and bottom right elements from the product of the top right and bottom left elements. So, if the matrix is [a b; c d], the determinant is ad - bc.
3. What is the significance of the determinant of a matrix?
Ans. The determinant of a matrix has several important applications in mathematics and physics. One of the most significant applications is in solving systems of linear equations. If the determinant of a matrix is zero, it means that the matrix is singular and the system of equations has no unique solution. If the determinant is non-zero, the system has a unique solution. Additionally, the determinant is used to find the inverse of a matrix, which is useful in many applications.
4. Can the determinant of a matrix be negative?
Ans. Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the order of the rows and columns of the matrix. Specifically, if you swap two rows or columns of a matrix, the sign of the determinant changes. If the number of row/column swaps is odd, the determinant is negative, and if it is even, the determinant is positive. So, even if all elements of a matrix are positive, the determinant can still be negative.
5. What is the relationship between the determinant of a matrix and its eigenvalues?
Ans. The determinant of a matrix is equal to the product of its eigenvalues. The eigenvalues of a matrix are the values of λ that satisfy the equation Ax = λx, where x is a non-zero vector. The determinant can be used to find the eigenvalues of a matrix, which is useful in many applications, including quantum mechanics and structural engineering. Additionally, the determinant can be used to determine whether a matrix is positive definite, which is important in optimization problems.
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