JEE Advanced (Assertion-Reason Type): Matrices & Determinants

Directions: The following questions consist of two statements, one labelled as "Assertion [A] and the other labelled as Reason [R]". You are to examine these two statements carefully and decide if Assertion [A] and Reason [R] are individually true and if so, whether the Reason [R] is the correct explanation for the given Assertion [A]. Select your answer from following options.
Q.1. Assertion [A]: is an identity matrix.
Reason [R]: A matrix A = [a_ij] is an identity matrix if
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (d)
We know that, is an indentity matrix
∴ Given Assertion [A] is false We know that for identity matrix  and
∴ Given Reason (R) is true Hence option [D] is the correct answer.

Q.2. Assertion [A]: Minor of element 6 in the matrix is 3.
Reason [R]: Minor of an element aij of a matrix is the determinant obtained by deleting its i^th row.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (e)
Minor of element  . ∴Given Assertion [A] is false Also we know that minor of an element aij of a matrix is the determinant obtained by deleting its i^th row and j^th column.
∴ Given Reason (R) is also false
∴ Both Assertion [A] and Reason (R) are false Hence option [E] is the correct Answer.

Q.3. Assertion [A]: Matrix  is a column matrix.
Reason [R]: A matrix of order m×1 is called a column matrix.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (a)
We know that order of column matrix is alway
 is column matrix.
Assertion [A] is true Also Reason (R) is true and is correct explanation of A.
Hence option [A] is the correct answer.

Q.4. Assertion [A]: For two matrices A and B of order 3, |A|=3,|B|=-4, then |2AB| is -96.
Reason [R]: For a matrix A of order n and a scalar k, |kA|=kⁿ|A|.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (b)
Here, |2AB|=2³|AB|=8|A||B| =8 × 3 × -4=-96
∴ Assertion [A] is true {∵ |kA| = kⁿ|A| and |AB|=|A||B|} Also we know that |kA| = kⁿ|A| for matrix A of order n.
∴ Reason (R) is true But |AB|=|A||B| is not mentioned in Reason R.
∴ Both A and R are true but R is not correct explanation of A
Hence option [B] is the correct answer.

Q.5. Assertion [A]: Transpose of the matrix A = [25 -1], is column matrix.
Reason [R]: Transpose of a matrix of order m × n is a matrix of same order.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (c)
Given A = [25-1]_1×3
, which is column matrix
∴ Given Assertion [A] is true We know that transpose of matrix of order m ×n is of order n×m
∴ Given Reason (R) is false Hence option [C] is the correct answer.

Q.6. Assertion (A): For a matrix  .
Reason (R): For  a  square  matrix  A,
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (a)
Given . We know A(adjA)A = |A| I  |
∴
∴ Both Assertion [A] and Reason (R) are true and R is the correct explanation of A.
Hence option [A] is the correct answer.

Q.7. Assertion [A]: For the matrix  
Reason [A]: For any matrix A, AI = IA = A.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (b)
We know that AB = I = BA ⇒ B = A^-1. Here   

 
 
 
⇒ Given Assertion [A] is true. Also AI = IA = A
⇒ Given Reason is true But given Reason is not correct explanation of A.
Hence option [B] is the correct answer.

Q.8. Assertion [A]: For a square matrix of order 2 A^-1 = 1/5adjA, so |2A| = 20.
Reason [R]: For a square matrix of order n, A^-1 =1 / |A|adjA and |adjA|=|A|^n-1.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (b)
Given  We know that
∴ |A| = 5 Also |2A| = 2²|A|  {∵|kA| = kⁿ|A|}  = 4 × 5 = 20
∴ Given Assertion [A] is true. Also Reason (R) is true Since |kA|=kn|A|is not mentioned in Reason
∴ Reason (R) is not correct explanation of A Hence option [B] is the correct answer.

Q.9. Assertion [A]: Matrix  is a skew symmetric matrix.
Reason [R]: A matrix A is skew symmetric if A′ = A.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false. 

Correct Answer is option (e)
Let
 
∴ Given Assertion [A] is false We know that A matrix A is skew symmetric if A′ = -A ∴Given Reason (R) is also false
∴ Both A and R are false
Hence option [E] is the correct answer.

Q.10. Assertion [A]: A^-1 exists
Reason [R]: R|A| = 0
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (c)
Given Assertion is A^-1 exists We know that
⇒ If |A| = 0, then A^-1 does not exist
∴ Reason |A| = 0is not valid for given Assertion
∴ A is true but R is false
Hence option [C] is the correct answer.

Q.11. Assertion [A]: Matrix  is a symmetric matrix.
Reason [R]: A matrix A is symmetric if A′ = -A.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (c)
Let

⇒ A is a symmetric matrix
∴ Given Assertion [A] is true We know that A matrix A is symmetric if A' = A
⇒ Given Reason is false A is true but R is false
Hence option [C] is the correct answer. 

Q.12. Assertion [A]: Value of x for which the matrix  is singular is -5.
Reason [R]: A matrix A is singular if|A| ≠ 0.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (c)
Given matrix A is singular 
∴
⇒  
⇒ 2(x + 4) - (0 - 2) = 0
⇒ 2x + 8 + 2 = 0
⇒2x = -10
⇒ x = -5
∴ Given Assertion [A] is true Also we know that For singular matrix A, |A|= 0
∴ Given Reason is false
Hence option [C] is the correct answer.

Q.13. Assertion [A]: For two matrices and  (A+B)² = A² + 2AB + B².
Reason [R]: For given two matrices A and B, AB = BA.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (a)
Given  
 
 
⇒ AB = BA. Now (A + B)² = (A + B)(A + B) = A²  + AB + BA + B2 = A² + AB + AB + B2 {∵AB = BA} = A² + 2AB + B² 
∴ Given Assertion [A] is true Also AB = BA
⇒ Reason (R) is true and is correct explanation of A
Hence option [A] is me correct answer.

Q.14. Assertion [A]: adj A is a non-singular matrix.
Reason [R]: A is non singular matrix.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (a)
Given Assertion adj A is non singular
⇒ |adjA| ≠ 0
We know that AadjA = |A|I_n
⇒ |AadjA| = |A|ⁿ|I_n|
⇒ |A||adjA| = |A|ⁿ.1
⇒ |adjA| = |A|^n-1
But |adjA| ≠ 0 ⇒|A|^n-1 ≠ 0
⇒ |A|≠0
⇒ Matrix A is non singular
⇒ R is true.
∴ Both A and R are true and R is the correct explanation of A
Hence option [A] is the correct answer.

Q.15. Assertion [A]: For a square matrix A, (2A)^-1 = 1 / 2A^-1.
Reason [R]: For any matrix A and scalar k, kA is a matrix obtained by multiplying each element of A by k.
(a) Both A and R are individually true and R is the correct explanation of A.
(b) Both A and R are individually true and R is not the correct explanation of A.
(c) 'A' is true but 'R' is false
(d) 'A' is false but 'R' is true
(e) Both A and R are false.

Correct Answer is option (d)
We know that,
∴ Given Assertion is not true Clearly Reason R is true ∴A is false and R is true Hence option [D] is the correct answer.