Assertion & Reason Type Questions: Determinants

Q1: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): If the matrix is singular, then λ = 4.

Reason (R): If A is a singular matrix, then |A|= 0.

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true but R is NOT the correct explanation of A

(c) A is true but R is false

(d) A is false and R is True

Ans: a

Sol: A matrix is said to be singular if |A| = 0

Hence R is true.

Hence A is true.

R is the correct explanation for A.

Q2: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Let A be a 2 × 2 matrix.

Assertion (A): adj (adj A) = A

Reason (R): |adj A|=|A|

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true but R is NOT the correct explanation of A

(c) A is true but R is false

(d) A is false and R is True

Ans: b

Sol: adj (adj A) = |A|n - 2 A

Here n = 2 ⇒ adj (adj A) = A

Hence A is true.

|adj A| = |A|^{n - 1} n = 2

⇒ |adj A|=|A|

Hence R is true.

R is not the correct explanation for A.

Q3: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): If every element of a third order determinant of value D is multiplied by 5, then the value of the new determinant is 125D.

Reason (R): If k is a scalar and A is an n × n matrix, then |kA|= kⁿ |A|

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true but R is NOT the correct explanation of A

(c) A is true but R is false

(d) A is false and R is True

Ans: a

Sol: If k is a scalar and A is an n × n matrix, then |kA| = kⁿ|A|.

This is a property of the determinant. Hence R is true.

Using this property, |5?| = 5³ ? = 125?

Hence A is true.

R is the correct explanation of A.

Q4: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): If and A^-1 = kA, then k = 1/9

Reason (R):

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true but R is NOT the correct explanation of A

(c) A is true but R is false

(d) A is false and R is True

Ans: d

Sol:

Q5: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): If then

Reason (R): The inverse of an invertible diagonal matrix is a diagonal matrix.

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true but R is NOT the correct explanation of A

(c) A is true but R is false

(d) A is false and R is True

Ans: a

Sol:

Q6: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Given

Assertion (A): 2A^-1 = 9I - A

Reason (R):

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true but R is NOT the correct explanation of A

(c) A is true but R is false

(d) A is false and R is True

Ans: a

Sol:

Hence R is true

∴ 2A^-1 =9I - A is true.

R is the correct explanation for A.