Class 12 RD Sharma Solutions - Chapter 6 Determinants Exercise Ex. 6.6 | Set 2

The Determinants are a fundamental concept in algebra and matrix theory widely used in solving systems of linear equations finding matrix inverses and analyzing the properties of the matrices. Understanding determinants is crucial for solving complex mathematical problems and provides a foundation for the higher mathematics and engineering disciplines.

Determinants

A determinant is a scalar value that can be computed from the square matrix. It provides the important properties of the matrix such as whether it is invertible and its volume scaling factor. The determinant of the matrix is denoted by the det(A) or ∣A∣. For a 2x2 matrix:

For larger matrices, determinants are computed using the more complex methods such as the cofactor expansion or row reduction.

Question 19. Let A = [a_ij] be a square matrix of order 3 × 3 and C_ij denotes the cofactor of a_ij in A. If |A| = 5, find the value of a₁₁ C₂₁ + a₁₂ C₂₂ + a₁₃ C₂₃.

Solution:

As we know that, if a matrix is square matrix of order n, then the sum of the products of elements of a row or a column with the cofactors of the corresponding elements of some other row or column is zero. 

So, 

A = [a_ij] is a square matrix of order n.

Also we have,

And 

=> a₁₁ C₂₁ + a₁₂ C₂₂ + a₁₃ C₂₃ = 0 

Therefore, the required value is 0.

Question 20. Find the value of .

Solution:

Given that, 

A = 

=> |A| = 

= sin 20° cos 70° + cos 20° sin 70°

= sin (20° + 70°)

= sin 90

= 1

Question 21. If A is a square matrix satisfying A^T A = I, write the value of |A|.

Solution:

Let us assume that A = [a_ij] be a square matrix of order n.

So, by using the property of determinants, we get

=> |A| = |A^T|

Here, we have

=> A^T A = I

=> |A^T A| = 1

So, the determinants are of same order, we get

=> |A^T A| = |A^T| |A|

=> |A^T| |A| = 1

=> 

=> 

=> |A|² = 1

=> |A| = ±1

Therefore, the value of |A| is ±1.

Question 22. If A and B are square matrices of the same order such that |A| = 3 and AB = I, then write the value of |B|.

Solution:

According to the question, A and B are square matrices of the same order.

So, by using the property of determinants we get,

=> |AB| = |A| |B|

Here, |A| = 3, AB = I.

=> |AB| = 1

=> |A| |B| = 1

=> 3 |B| = 1

=> |B| = 1/3

Therefore, the value of |B| is 1/3.

Question 23. A is a skew-symmetric of order 3, write the value of |A|.

Solution:

Here, |A| = 4.

So we have,

Order of the matrix (n) = 3

Using the properties of matrices, we get

For a square matrix of order n and constant k, we know,

=> |k A| = kⁿ |A|

=> |- A| = (-1)³ |A|

= (-1) (4)

= -4

Therefore, the value of |A| is -4.

Question 24. If A is a square matrix of order 3 with determinant 4, then write the value of |−A|.

Solution:

Given that, |A| = 4.

Order of the matrix (n) = 3

So, by using the properties of matrices, we get

=> |k A| = kⁿ |A|

=> |- A| = (-1)³ |A|

= (-1) (4)

= -4

Therefore, the value of |A| is -4.

Question 25. If A is a square matrix such that |A| = 2, write the value of |A A^T|.

Solution:

Given that, |A| = 2

As we know that in a square matrix, |A| = A^T

So, they are of sane order 

Hence, |A A^T| = |A| |A^T|

=> |A A^T| = 2 (2)

= 4

Therefore, the value of |A A^T| is 4.

Question 26. Find the value of the determinant .

Solution:

Given that, 

A = 

|A| = 

On applying R₁ -> R₁ - 3R₂ we have,

= 

= 

= 0

Therefore, the value of the determinant is 0.

Question 27. Find the value of the determinant .

Solution:

Given that, 

A = 

|A| = 

On applying R₂ -> R₂ - 2R₁ we get,

= 

= 

= 0

Therefore, the value of the determinant is 0.

Question 28. If the matrix  is singular, find the value of x.

Solution:

As we know that a matrix is singular only when its determinant is zero. 

According to the question, 

 is a singular matrix

So, 

=> |A| =  = 0

On expanding the determinant we get,

=> 5x + 20 = 0 

=> x = -20/5

=> x = -4

Therefore, the value of x is -4.

Question 29. If A is a square matrix of order n × n such that |A| = λ, then write the value of |−A|.

Solution:

Given that,

A is a square matrix of order n × n

So, |A| = λ

=> |- A| = (-1)ⁿ A

=> |-A| = (-1)ⁿ λ

Therefore, the value of |-A| is (-1)ⁿ λ.

Question 30. Find the value of the determinant .

Solution:

Given that, 

A = 

|A| = 

On taking out common factors from R₁, R₂ and R₃ we get,

= 

Here, the two rows are identical, so we get

= 

= 0 

Therefore, the value of the determinant is 0.

Question 31. If A and B are non-singular matrices of the same order, prove whether AB is singular or non-singular.

Solution:

According to the question, A and B be non-singular matrices of order n.  

Here, |A| ≠ 0 and |B| ≠ 0. 

So, the order of these matrix are same, we get

=> |AB| = |A| |B|

=> |AB| = 0 if |A| = 0 or |B| = 0

But as it is not the case here, so |AB| is non- zero matrix and AB is non-singular matrix.

Hence proved.

Question 32. A matrix of order 3 × 3 has determinant 2. What is the value of |A (3I)|, where I is the identity matrix of order 3 × 3.

Solution:

Given that a matrix of order 3 x 3 has determinant 2. 

So let us assume B be the matrix. so the order of the matrix is 3

and |B| = 2  

Let us consider I be the identity matrix, so we get

=> |I| = 1 

=> 3 |I| = 3

=> |A (3I)| = |3 A|

= 3³ |A|

= 27 (2)

= 54

=> |A (3I)| = 54

Therefore, the value of |A (3I)| is 54.

Question 33. If A and B are square matrices of order 3 such that |A| = −1, |B| = 3, then find the value of |3 AB|.

Solution:

We have,

A and B are square matrices of order 3.

Also |A| = −1, |B| = 3.

We know,

As n is the order of A, we get

=> |K A| = Kⁿ |A| 

=> |3 AB| = 3³ |AB|  

If the order of A and B matrix are same and they are square matrix then |AB| = |A| |B|. 

So, we have,

=> |3 AB| = 3³ |A| |B|

= 27 (-1) (3)

= -81

Therefore, the value of |3 AB| is -81.

Question 34. Write the value of .

Solution:

We have,

A = 

|A| = 

= a² - iab + iab - i² b² - (-c² - icd + icd + i² d²) 

= a² - i² b² + c² - i² d²

Here, we have, i² = - 1.

So we get,  

|A| = a² - (-1) b² + c² - (-1) d²

= a² + b² + c² + d²

Therefore, the value is a² + b² + c² + d².

Question 35. Write the cofactor of a₁₂ in the following matrix .

Solution:

We have

So,

=> a₁₂ = -3

Now we find the cofactor of a₁₂

a₁₂ = (-1)¹⁺²

= - (- 42 - 4) 

= 46

Therefore, the value of the required cofactor is 46.

Question 36. If , find x.

Solution:

Here we have,

A = 

=> |A| =  = 0

=> 9(2x + 5) - 3(5x + 2) = 0

=> 18x + 45 - 15x - 6 = 0  

=> 3x + 39 = 0

=> 3x = - 39 

=> x = -39/3 

=> x = -13

Therefore, the value of x is -13.

Question 37. Find the value of x from the following : 

Solution:

We have,

A = 

|A| = 

=>  = 0 

=> 2 x² - 8 = 0 

=> 2 x² = 8

=> x² = 8/2

=> x² = 4  

=> x = √4

=> x = ±2

Therefore, the value of x is ±2.

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