If $A$ is square matrix, then

${A^T} + A\;$ is skew-symmetric

$\left| {\begin{array}{*{20}{c}} {{{\sin }^2}x}&{{{\cos }^2}x}&1\\ {{{\cos }^2}x}&{{{\sin }^2}x}&1\\ { - 10}&{12}&2 \end{array}} \right| = $

$12{\sin ^2}x - 10{\cos ^2}x - 2$

If $n$ is a natural number. Then ${\left[ {\begin{array}{*{20}{c}} 2&{ - 1}\\ 3&{ - 2} \end{array}} \right]^n},$ is

$\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]$ is $n$ is even

$\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]$ if $n$ is odd

$\left[ {\begin{array}{*{20}{c}} { - 1}&0\\ 0&1 \end{array}} \right]$ if $n$ is a natural number

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JEE • MATHEMATICS • MATRICES

Matrices practice · Easy

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 601-630 of 922 questions

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601 Easy

601 of 922

Understanding +4 | -1

The trace

T_r(A)

of a

3 \times 3

matrix

A=\left(a_{i j}\right)

is defined by the relation

T_r(A)=a_{11}+a_{22}+a_{33}

(i.e.

T_r(A)

is sum of the main diagonal elements). Which of the following statements cannot hold ?

View answer and explanation

Correct option: D

\mathrm{T}_{\mathrm{r}}\left(\mathrm{A}^2\right)=\mathrm{T}_{\mathrm{r}}(\mathrm{A})^2

Option D (

\mathrm{T}_{\mathrm{r}}(\mathrm{A}^2)=\mathrm{T}_{\mathrm{r}}(\mathrm{A})^2

) cannot hold because the trace of A squared is not generally equal to the square of the trace of A.

602 Easy

602 of 922

Analyzing +4 | -1

\Delta=\left|\begin{array}{ccc}0 & b-a & c-a \\ a-b & 0 & c-b \\ a-c & b-c & 0\end{array}\right|,

then

\Delta

equals

View answer and explanation

Correct option: D

Option D (0) is correct because the given determinant represents a skew-symmetric matrix of odd order, and the determinant of any skew-symmetric matrix of odd order is always zero.

603 Easy

603 of 922

Application +4 | -1

A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right]

and

B=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]

, then

A^8

equals

View answer and explanation

Correct option: A

128 {~B}

Option A (

128 {~B}

) is correct because after repeated squaring of matrix A, it simplifies to a multiple of matrix B.

604 Easy

604 of 922

Understanding +4 | -1

\mathrm{B}

is a non-singular matrix and

\mathrm{A}

is a square matrix, then

\operatorname{det}\left(\mathrm{B}^{-1} \mathrm{AB}\right)

is equal to

View answer and explanation

Correct option: C

\operatorname{det}(\mathrm{A})

Option C (det(A)) is correct because the determinant of B⁻¹AB simplifies to the determinant of A using properties of determinants.

605 Easy

605 of 922

Application +4 | -1

Let

P=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{array}\right]

and

\mathrm{I}

be the identity matrix of order 3. If

Q=\left[q_{i j}\right]

is a matrix such that

p^{50}-Q

is I, then

\frac{q_{31}+q_{32}}{q_{21}}

equals to

View answer and explanation

Correct option: D

103

Option D (103) is correct. The matrix P raised to the power of 50 is calculated, and then the elements of matrix Q are found by subtracting I from P⁵⁰, and finally, the expression is evaluated.

606 Easy

606 of 922

Analyzing +4 | -1

Let

M = {\left[ {{a_{uv}}} \right]_{n \times n}}

be a matrix, where

{a_{uv}} = \sin \left( {{\mathrm{\theta }_u} - {\mathrm{\theta }_v}} \right) + i\cos \left( {{\mathrm{\theta }_u} - {\mathrm{\theta }_v}} \right)

, the

M

is equal to

View answer and explanation

Correct option: D

- {\bar M^T}

Option D (

-{\bar M^T}

) is correct because taking the conjugate transpose of the conjugate of M results in the negative of the original matrix M.

607 Easy

607 of 922

Application +4 | -1

A, B

and

C

are three square matrices of the same size such that

B=C A C^{-1}

, then

C A^3 C^{-1}

is equal to

View answer and explanation

Correct option: C

B^3

Option C (

B^3

) is correct because by repeatedly substituting B = CAC^-1, we can show that CA³C^-1 = B³.

608 Easy

608 of 922

Application +4 | -1

A = \left[ {\begin{array}{cccccccccccccccccccc} {0\;\;\;\;\;3\;\;\;\;\;\;3}\\ { - 3\;\;\;\;0\; - 4}\\ { - 3\;\;\;\;4\;\;\;\;\;\;0} \end{array}} \right]

and

\;B = \left[ {\begin{array}{cccccccccccccccccccc} x\\ y\\ z \end{array}} \right],\;

then

\;B'\left( {AB} \right)

View answer and explanation

Correct option: A

Null matrix

Option A (Null matrix) is correct because the resulting matrix is a 1x1 matrix with a value of 0, making it a null matrix.

609 Easy

609 of 922

Analyzing +4 | -1

Given,

f(x)=\left|\begin{array}{ccc}0 & x^2-\sin x & \cos x-2 \\ \sin x-x^2 & 0 & 1-2 x \\ 2-\cos x & 2 x-1 & 0\end{array}\right|

, then

\int f(x) d x

is equal to

View answer and explanation

Correct option: D

None of these

The correct answer is None of these because the determinant of the given matrix is zero, leading to the integral of zero, which is a constant.

610 Easy

610 of 922

Understanding +4 | -1

A

is a square matrix such that

A^3=I

then

A^{-1}

is equal to

View answer and explanation

Correct option: C

A^2

Option C (

A^2

) is correct because if

A^3 = I

, then multiplying both sides by

A^{-1}

gives

A^2 = A^{-1}

611 Easy

611 of 922

Application +4 | -1

A=\left[\begin{array}{cc}\frac{-1+i \sqrt{3}}{2 i} & \frac{-1-i \sqrt{3}}{2 i} \\ \frac{1+i \sqrt{3}}{2 i} & \frac{1-i \sqrt{3}}{2 i}\end{array}\right], i=\sqrt{-1}

and

f(x)=x^2+2,

then

f(A)

equals to

View answer and explanation

Correct option: D

(2+i \sqrt{3})\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

Option D ((2+i√3)[1 0
0 1]) is correct because substituting the values and simplifying leads to f(A) = (2+i√3)I.

612 Easy

612 of 922

Application +4 | -1

The simultaneous equations

Kx + 2y - z = 1,\left( {K - 1} \right)y - 2z = 2\;

and

\;\left( {K + 2} \right)z = 3\;

have only one solution when

View answer and explanation

Correct option: B

K = - 1

Option B (K = -1) is correct because for the system to have a unique solution, the determinant of the coefficient matrix must be non-zero, which holds true when K = -1.

613 Easy

613 of 922

Understanding +4 | -1

A

is square matrix, then

View answer and explanation

Correct option: A

A + {A^T}

is symmetric

Option A is correct because the transpose of the sum of a matrix and its transpose is equal to the sum itself, indicating symmetry.

614 Easy

614 of 922

Understanding +4 | -1

Which of the following is correct statement ?

View answer and explanation

Correct option: B

Identity matrix is a particular case of scalar matrix

Option B is the correct statement because an identity matrix is a special case of a scalar matrix where all diagonal elements are equal to 1.

615 Easy

615 of 922

Application +4 | -1

A

is a square matrix or order

n \times n,

then

\mathrm{adj}\left( {\mathrm{adj\;}A} \right)

is equal to

View answer and explanation

Correct option: C

{\left| A \right|^{n - 2}}A

Option C (

{\left| A \right|^{n - 2}}A

) is correct because it is a standard result for the adjoint of the adjoint of a matrix.

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616 Easy

616 of 922

Application +4 | -1

A

is a square matrix, then

\mathrm{adj\;}{A^T} - {\left( {\mathrm{adj\;}A} \right)^T}

is equal to

View answer and explanation

Correct option: C

Null matrix

Option C (Null matrix) is correct because the transpose of the adjoint of a matrix is equal to the adjoint of the transpose of the matrix.

617 Easy

617 of 922

Analyzing +4 | -1

The number of

3 \times 3

non-singular matrices, with four entries as 1 and all other entries as 0, is

View answer and explanation

Correct option: C

Atleast 7

Option C (Atleast 7) is correct because there are at least 6 ways to place the four 1s such that each row and column has at least one 1, and other configurations also exist.

618 Easy

618 of 922

Analyzing +4 | -1

O(A)=2 \times 3, O(B)=3 \times 2

, and

O(C)=3 \times 3

, which one of the following is not defined

View answer and explanation

Correct option: D

\mathrm{C}\left(\mathrm{A}+\mathrm{B}^r\right)

Option D (

\mathrm{C}\left(\mathrm{A}+\mathrm{B}^r\right)

) is not defined because the dimensions of the matrices A and B' do not allow for addition, and even if they did, the resulting matrix could not be multiplied by C due to incompatible dimensions.

619 Easy

619 of 922

Understanding +4 | -1

Let

A=\left(a_{i j}\right)_{3 \times 2}

be a

3 \times 2

matrix with real entries and

B=A A^{\prime}

. Then

View answer and explanation

Correct option: C

B^{-1}

does not exist

Option C (

B^{-1}

does not exist) is correct because the determinant of B is always zero, making B a singular matrix and thus non-invertible.

620 Easy

620 of 922

Application +4 | -1

P=\left[\begin{array}{lll}1 & k & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]

is the adjoint of a

3 \times 3

matrix

Q

and

\operatorname{det}(Q)=6

, then

k

is equal to

View answer and explanation

Correct option: C

Option C (36) is correct. The determinant of the adjoint of a matrix Q is equal to the determinant of Q squared.

621 Easy

621 of 922

Analyzing +4 | -1

\left| {\begin{array}{cccccccccccccccccccc} {{{\sin }^2}x}&{{{\cos }^2}x}&1\\ {{{\cos }^2}x}&{{{\sin }^2}x}&1\\ { - 10}&{12}&2 \end{array}} \right| =

View answer and explanation

Correct option: A

Option A (0) is correct because after applying column operations, two columns of the determinant become identical, resulting in a zero determinant.

622 Easy

622 of 922

Analyzing +4 | -1

If the system of equations

x+y+z=6

\,\,\,\,\,2 x+5 y+\alpha z=\beta,\,\,\,

x+2 y+3 z=14

has infinitely many solutions, then

\alpha + \beta

is equal to

View answer and explanation

Correct option: C

Option C (44) is correct because it is the sum of

\alpha

and

\beta

when the system of equations has infinitely many solutions, found by setting the determinant to zero and solving for the parameters.

623 Easy

623 of 922

Analyzing +4 | -1

Let

P=\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & a \\ 3 & -5 & 0\end{array}\right]

where

a \in R

, suppose

Q=\left[q_{i j}\right]_{3 \times 3}

Matrix such that

P Q=\mathrm{KI}_3

for some non-zero

K \in R

.If

q_{23}=\frac{-k}{8}

and

|Q|=\frac{k^2}{2}

, then

a^2+k^2

View answer and explanation

Correct option: B

Prime number

Option B (Prime number) is correct because after solving the equations derived from the given conditions, we find a = -1 and k = 4, so a² + k² = 17, which is a prime number.

624 Easy

624 of 922

Application +4 | -1

n

is a natural number. Then

{\left[ {\begin{array}{cccccccccccccccccccc} 2&{ - 1}\\ 3&{ - 2} \end{array}} \right]^n},

View answer and explanation

Correct option: A

\left[ {\begin{array}{cccccccccccccccccccc} 1&0\\ 0&1 \end{array}} \right]

n

is even

Option A (

\left[ {\begin{array}{cccccccccccccccccccc} 1&0\\ 0&1 \end{array}} \right]

n

is even) is correct because when

n

is even, the matrix raised to the power of

n

results in the identity matrix.

625 Easy

625 of 922

Application +4 | -1

\mathrm{A}=\left[\begin{array}{cc}\sin \alpha & -\cos \alpha \\ \cos \alpha & \sin \alpha\end{array}\right]

and

\mathrm{A}+\mathrm{A}^{\prime}=\mathrm{I}

, then

\alpha=? \quad (0 < \alpha < \frac{\pi}{2})

View answer and explanation

Correct option: B

\frac{\pi}{6}

Option B is correct. By adding A and A', and setting the result equal to I, we can solve for alpha and find that it equals pi/6.

626 Easy

626 of 922

Application +4 | -1

Let

D=\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|; 0 \leq \theta<2 \pi,

then

View answer and explanation

Correct option: C

D \in[2,4]

Option C (D ∈ [2,4]) is correct because the determinant simplifies to 2(1 + sin²(theta)), and since 0 <= sin²(theta) <= 1, then 2 <= D <= 4.

627 Easy

627 of 922

Application +4 | -1

A

is a non-singular matrix of order 3, then

adj(adj\,A)

is equal to

View answer and explanation

Correct option: D

\left| A \right|A

Option D (

\left| A \right|A

) is correct because for a non-singular matrix

A

of order 3,

adj(adj A) = |A|A

628 Easy

628 of 922

Application +4 | -1

A=\left[\begin{array}{rr}2 & -3 \\ -4 & 1\end{array}\right]

, then adj

\left(3 A^2+12 A\right)

is equal to

View answer and explanation

Correct option: B

\left[\begin{array}{ll}51 & 63 \\ 84 & 72\end{array}\right]

Option B (

\left[\begin{array}{ll}51 & 63 \\ 84 & 72\end{array}\right]

) is correct because after calculating 3A² + 12A, the adjugate of the resulting matrix is the given option.

629 Easy

629 of 922

Understanding +4 | -1

\mathrm{A}

and

\mathrm{B}

are matrices of same order, then

\left(\mathrm{AB}^{\prime}-\mathrm{BA}^{\prime}\right)

is a

View answer and explanation

Correct option: A

skew symmetric matrix

Option A (skew symmetric matrix) is correct because (AB' - BA')' = -(AB' - BA'), which satisfies the definition of a skew-symmetric matrix.

630 Easy

630 of 922

Application +4 | -1

A

is a

3 \times 3

matrix and

|A|=2

, then

\left|3 \operatorname{adj}\left(|3 A| A^2\right)\right|

is equal to

View answer and explanation

Correct option: D

3^{11} \cdot 6^{10}

Option D (

3^{11} \cdot 6^{10}

) is correct because it involves using properties of determinants and adjoints to simplify the expression.

Revision notes and key takeaways

Review MATRICES in your textbook or class notes, then use this set to check your understanding.

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