JEE Main > Matrices and Determinants

Explore popular questions from Matrices and Determinants for JEE Main. This collection covers Matrices and Determinants previous year JEE Main questions hand picked by popular teachers.


Q 1.    

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { c c c } { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { a } & { b } & { - 1 } \end{array} \right] , {/tex} then {tex} A ^ { 2 } {/tex} is equal to

Unit matrix

B

Null matrix

C

{tex} A {/tex}

D

{tex} - A {/tex}

Explanation

Q 2.    

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { l l l } { a } & { 0 } & { 0 } \\ { 0 } & { a } & { 0 } \\ { 0 } & { 0 } & { a } \end{array} \right] , {/tex} then {tex} | A | | {/tex} adj {tex} A | {/tex} is equal to

A

{tex} a ^ { 25 } {/tex}

B

{tex} a ^ { 27 } {/tex}

C

{tex} a ^ { 81 } {/tex}

None of these

Explanation

Q 3.    

Correct4

Incorrect-1

If {tex} A {/tex} is {tex} a \,\,3 \times 3 {/tex} skew-symmetric matrix, then {tex} | A | {/tex} is given by

0

B

{tex} - 1 {/tex}

C

1

D

None of these

Explanation

Q 4.    

Correct4

Incorrect-1

If {tex} A {/tex} and {tex} B {/tex} are two square matrices of the same order and {tex} m {/tex} is a positive integer, then
{tex} ( A + B ) ^ { m } = ^ { m } C _ { 0 } A ^ { m } + ^ { m } C _ { 1 } A ^ { m - 1 } B + ^ { m } C _ { 2 } A ^ { m - 2 } B ^ { 2 } + \cdots + ^ { m } C ^ { m - 1 } A B ^ { m - 1 } {/tex} {tex} + ^ { m } C _ { m } B ^ { m } {/tex} if

{tex} A B = B A {/tex}

B

{tex} A B + B A = 0 {/tex}

C

{tex} A ^ { m } = 0 , B ^ { m } = 0 {/tex}

D

None of these

Explanation



Q 5.    

Correct4

Incorrect-1

If {tex} A = \left( a _ { i j } \right) _ { 3 \times 3 } {/tex} is a skew-symmetric matrix, then

{tex} a _ { i i } = 0 , \forall i {/tex}

B

{tex} A - A ^ { \prime } {/tex} is null matrix

C

{tex} | A | \neq 0 {/tex}

D

None of these

Explanation





Q 6.    

Correct4

Incorrect-1

{tex} \left[ \begin{array} { l l l } { 7 } & { 1 } & { 2 } \\ { 9 } & { 2 } & { 1 } \end{array} \right] \left[ \begin{array} { l } { 3 } \\ { 4 } \\ { 5 } \end{array} \right] + 2 \left[ \begin{array} { l } { 4 } \\ { 2 } \end{array} \right] {/tex} is equal to

{tex} \left[ \begin{array} { c } { 43 } \\ { 44 } \end{array} \right] {/tex}

B

{tex} \left[ \begin{array} { l } { 43 } \\ { 45 } \end{array} \right] {/tex}

C

{tex} \left[ \begin{array} { c } { 45 } \\ { 44 } \end{array} \right] {/tex}

D

None of these

Explanation

Q 7.    

Correct4

Incorrect-1

For any matrix {tex} A {/tex} of order {tex} 2 \times 2 , {/tex} if {tex} A ( \text { adj } A ) = \left[ \begin{array} { c c } { 10 } & { 0 } \\ { 0 } & { 10 } \end{array} \right] , {/tex} then

A

20

B

100

10

D

0

Explanation

Q 8.    

Correct4

Incorrect-1

If a matrix {tex} A {/tex} is symmetric as well as skew-symmetric, then {tex} A {/tex} is a

A

Diagonal matrix

Null matrix

C

Unit matrix

D

None of these

Explanation

Q 9.    

Correct4

Incorrect-1

If {tex} A {/tex} and {tex} B {/tex} be two square matrices such that {tex} A B = O {/tex} , then

{tex} \operatorname { det } A = 0 {/tex} or det {tex} B = 0 {/tex}

B

{tex} \operatorname { det } B = 0 {/tex}

C

{tex} B = A ^ { - 1 } {/tex}

D

{tex} \operatorname { det } A = 0 {/tex}

Explanation

Q 10.    

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { l l l } { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \end{array} \right] , {/tex} then

A

{tex} A ^ { 2 } = A {/tex}

B

{tex} A ^ { 2 } = 0 {/tex}

C

{tex} A ^ { 2 } = l {/tex}

{tex} A ^ { 3 } = O {/tex}

Explanation

Q 11.    

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { l l l } { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \end{array} \right] , {/tex} then {tex} A {/tex} is

A

An invertible matrix

B

An idempotent matrix

A nilpotent matrix

D

None of these

Explanation



Q 12.    

Correct4

Incorrect-1

If {tex} A {/tex} and {tex} B {/tex} are symmetric matrices of the same order, then

A

{tex} A B {/tex} is a symmetric matrix

B

{tex} A - B {/tex} is skew-symmetric matrix

{tex} A B + B A {/tex} is a symmetric matrix

D

{tex} A B - B A {/tex} is a symmetric matrix

Explanation





Q 13.    

Correct4

Incorrect-1

If {tex} A {/tex} is any square matrix, then

A

{tex} A + A ^ { \prime } {/tex} is skew-symmetric

B

{tex} A - A ^ { \prime } {/tex} is symmetric

{tex} A A ^ { \prime } {/tex} is symmetric

D

None of these

Explanation

Q 14.    

Correct4

Incorrect-1

If {tex} A {/tex} is a square matrix such that {tex} A ^ { 3 } = l {/tex} then {tex} A ^ { - 1 } {/tex} is equal to

A

l

B

{tex} A {/tex}

{tex} A ^ { 2 } {/tex}

D

None of these

Explanation

Q 15.    

Correct4

Incorrect-1

If {tex} A {/tex} is any square matrix then which of the following is not symmetric?

A

{tex} A + A ^ { \prime } {/tex}

{tex} A - A ^ { \prime } {/tex}

C

{tex} A A ^ { \prime } {/tex}

D

{tex} A ^ { \prime } A {/tex}

Explanation

Q 16.    

Correct4

Incorrect-1

Let {tex} A {/tex} be a skew-symmetric matrix of order {tex} n {/tex} then

A

{tex} | A | = 0 {/tex} if {tex} n {/tex} is even

{tex} | A | = 0 {/tex} if {tex} n {/tex} is odd

C

{tex} | A | = 0 {/tex} for all {tex} n \in N {/tex}

D

None of these

Explanation

Q 17.    

Correct4

Incorrect-1

Each diagonal element of skew-symmetric matrix is

Zero

B

Positive

C

Non-real

D

Negative

Explanation

Q 18.    

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { c c } { 4 } & { 2 } \\ { - 1 } & { 1 } \end{array} \right] , {/tex} then the value of {tex} ( A - 2 n ) ( A - 3 n \text { is } {/tex}

A

Unit matrix

B

Non-singular matrix

Null matrix

D

None of these

Explanation



Q 19.    

Correct4

Incorrect-1

Matrix {tex} A {/tex} has {tex} m {/tex} rows and {tex} n + 5 {/tex} columns, matrix {tex} B {/tex} has {tex} m {/tex} rows and {tex} 11 - n {/tex} columns. If both {tex} A B {/tex} and {tex} B A {/tex} exist, then

{tex} A B {/tex} and {tex} B A {/tex} are square matrices

B

{tex} A B {/tex} and {tex} B A {/tex} are of order {tex} 8 \times 8 {/tex} and {tex} 3 \times 13 , {/tex} respectively

C

{tex} A B = B A {/tex}

D

None of these

Explanation



Q 20.    

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { l l l } { 1 } & { 3 } & { 3 } \\ { 1 } & { 4 } & { 3 } \\ { 1 } & { 3 } & { 4 } \end{array} \right] , {/tex} then {tex} A ^ { - 1 } {/tex} is equal to

A

{tex} \left[ \begin{array} { c c c } { 7 } & { - 3 } & { - 3 } \\ { 0 } & { 1 } & { 0 } \\ { - 1 } & { 0 } & { 5 } \end{array} \right] {/tex}

{tex} \left[ \begin{array} { c c c } { 7 } & { - 3 } & { - 3 } \\ { - 1 } & { 1 } & { 0 } \\ { - 1 } & { 0 } & { 1 } \end{array} \right] {/tex}

C

{tex} \left[ \begin{array} { c c c } { 7 } & { - 3 } & { - 3 } \\ { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 1 } \end{array} \right] {/tex}

D

None of these

Explanation







Q 21.    

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { c c } { i } & { - i } \\ { - i } & { i } \end{array} \right] {/tex} and {tex} B = \left[ \begin{array} { c c } { 1 } & { - 1 } \\ { - 1 } & { 1 } \end{array} \right] , {/tex} then {tex} A ^ { 8 } {/tex} equals

128{tex} B {/tex}

B

132{tex} B {/tex}

C

116{tex} B {/tex}

D

8{tex} B {/tex}

Explanation



Q 22.    

Correct4

Incorrect-1

If {tex} \alpha , \beta , \gamma {/tex} are three real numbers and
{tex} A = \left[ \begin{array} { c c c } { 1 } & { \cos ( \alpha - \beta ) } & { \cos ( \alpha - \gamma ) } \\ { \cos ( \beta - \alpha ) } & { 1 } & { \cos ( \beta - \gamma ) } \\ { \cos ( \gamma - \alpha ) } & { \cos ( \gamma - \beta ) } & { 1 } \end{array} \right] {/tex} then

A

{tex} A {/tex} is skew-symmetric

B

{tex} A {/tex} is invertible

C

{tex} A {/tex} is non-singular

{tex} | A | = 0 {/tex}

Explanation









Q 23.    

Correct4

Incorrect-1

If {tex} \left| \begin{array} { l l l } { a _ { 1 } } & { b _ { 1 } } & { c _ { 1 } } \\ { a _ { 2 } } & { b _ { 2 } } & { c _ { 2 } } \\ { a _ { 3 } } & { b _ { 3 } } & { c _ { 3 } } \end{array} \right| \neq 0 , {/tex} then the number of solutions of the
system of equations {tex} a _ { 1 } x + b _ { 1 } y + c _ { 1 } z = 0 , a _ { 2 } x + b _ { 2 } y + c _ { 2 } z = 0 {/tex} and {tex} a _ { 3 } x + b _ { 3 } y + c _ { 3 } z = 0 {/tex} is

A

Infinite number of solutions

Only one unique solution

C

More than one solution

D

None of these

Explanation

Q 24.    

Correct4

Incorrect-1

Let {tex} A {/tex} and {tex} B {/tex} be two square matrices of the same dimension and let {tex} [ A , B ] = A B - B A {/tex} . Then for three {tex} 2 \times 2 {/tex} matrices {tex} A , B , C , {/tex}
{tex} [ [ A , B ] , C ] + [ [ B , C ] , A ] + [ [ C , A ] , B ] {/tex} is equal to

A

1

0

C

{tex} A B C - C B A {/tex}

D

None of these

Explanation



Q 25.    

Correct4

Incorrect-1

If the matrices {tex} A , B , ( A + B ) {/tex} are non-singular, then
{tex} \left[ A ( A + B ) ^ { - 1 } B \right] ^ { - 1 } {/tex} is equal to

A

{tex} A + B {/tex}

B

{tex} A ^ { - 1 } + B ^ { - 1 } {/tex}

C

{tex} ( A + B ) ^ { - 1 } {/tex}

None of these

Explanation