The inverse matrix of $A = \left[ {\begin{array}{*{20}{c}} {0\;\;1\;\;2}\\ {1\;\;\,2\;\;3}\\ {3\;\;1\;\;1} \end{array}} \right]$ is

$\left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&{ - \frac{1}{2}}&{\frac{1}{2}}\\ { - 4}&3&{ - 1}\\ {\frac{5}{2}}&{ - \frac{3}{2}}&{\frac{1}{2}} \end{array}} \right]$

$\left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&{ - 4}&{\frac{5}{2}}\\ 1&{ - 6}&3\\ 1&2&{ - 1} \end{array}} \right]$

$\frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1&2&3\\ 3&2&1\\ 4&2&3 \end{array}} \right]$

$\frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&{ - 1}\\ { - 8}&6&{ - 2}\\ 5&{ - 3}&1 \end{array}} \right]$

$\left| {\begin{array}{*{20}{c}} {\rm{\alpha }}&{ - {\rm{\beta }}}&0\\ 0&{\rm{\alpha }}&{\rm{\beta }}\\ {\rm{\beta }}&0&{\rm{\alpha }} \end{array}} \right| = 0$, then

${\rm{\alpha }}$ is one of the cube roots of unity

${\rm{\alpha \beta }}$ is one of the cube roots of unity

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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 901-922 of 922 questions

901 Easy

901 of 922

Analyzing +4 | -1

A B=A

and

B A=B

, where

A

and

B

are square matrices, then

View answer and explanation

Correct option: A

B^2=B

and

A^2=A

The correct answer is

B^2=B

and

A^2=A

because given

AB=A

and

BA=B

, it can be shown that

A^2 = A

and

B^2 = B

902 Easy

902 of 922

Analyzing +4 | -1

The number of values of

k

for which the system of equations

(k+1) x+8 y=4 k ; k x+(k+3) y=3 k-1

has no solution, is

View answer and explanation

Correct option: A

Option A (1) is correct because the system has no solution when the determinant of the coefficients is zero and at least one of the other determinants is non-zero, which occurs for one value of

k

903 Easy

903 of 922

Understanding +4 | -1

A

and

B

are two matrices such that both

A + B

and

AB

are defined, then

View answer and explanation

Correct option: C

Both

A

and

B

are of same order

n \times n

Option C (Both

A

and

B

are of same order

n \times n

) is correct because both

A + B

and

AB

being defined requires

A

and

B

to be square matrices of the same order.

904 Easy

904 of 922

Application +4 | -1

Let

A=\left[a_{i y}\right]_{3 \times 3}

be such that

a_{i j}=\left[\begin{array}{ll}3 \text {; when } & \hat{i}=\hat{j} \\ 0 ; & \hat{i} \neq \hat{j}\end{array}\right.

, then

\left\{\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj} A))}{5}\right\}

equals: (where

\{\cdot\}

denotes fractional part function)

View answer and explanation

Correct option: A

0.2

Option B (1) is correct because the expression simplifies to sin(π/2) + cos(π/2) which equals 1.

905 Easy

905 of 922

Application +4 | -1

\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]-\left[\begin{array}{lr}4 & -2 \\ 0 & -6 \\ -1 & 2\end{array}\right]\left[\begin{array}{l}2 \\ 1\end{array}\right]

then

(x, y, z)

is equal to -

View answer and explanation

Correct option: A

(-4,2,2)

Option A ((-4,2,2)) is correct because it is the solution to the system of linear equations obtained by matrix multiplication and simplification.

906 Easy

906 of 922

Application +4 | -1

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

and

n \in N,

then

{A^n}

is equal to

View answer and explanation

Correct option: B

{2^{n - 1}}\;A

Option B is correct. By observing the pattern of A raised to successive powers, it can be generalized that A^n = 2^(n-1) * A.

907 Easy

907 of 922

Application +4 | -1

Let

A=\left[\begin{array}{lll}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]

and suppose that

\operatorname{det}(A)=2

, then

\operatorname{det}(B)

equals, where

B=\left[\begin{array}{ccc}4 x & 2 a & -p \\ 4 y & 2 b & -q \\ 4 z & 2 c & -r\end{array}\right]

View answer and explanation

Correct option: C

-16

Option C (-16) is correct because applying determinant properties such as row swapping and scalar multiplication leads to det(B) = -16.

908 Easy

908 of 922

Analyzing +4 | -1

If the system of linear equations

x + y + z = 5,\,\,\,x + 2y + 2z = 6,\,\,\,x + 3y + \lambda z = \mu ,\,\,\,\,\left( {\lambda ,\mu \in R} \right),

has infinitely many solutions, then the value of

\lambda + \mu

View answer and explanation

Correct option: C

Option C (10) is correct because for the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the equations must be consistent, leading to

\lambda = 3

and

\mu = 7

, so

\lambda + \mu = 10

909 Easy

909 of 922

Application +4 | -1

A=\left[\begin{array}{ccc}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right]

and

B=\left[\begin{array}{ccc}a^2 & a b & a c \\ a b & b^2 & b c \\ a c & b c & c^2\end{array}\right]

, then

A B

is equal to

View answer and explanation

Correct option: C

O

Option C (

O

) is correct because the product of matrices A and B results in a zero matrix.

910 Easy

910 of 922

Application +4 | -1

The inverse matrix of

A = \left[ {\begin{array}{cccccccccccccccccccc} {0\;\;1\;\;2}\\ {1\;\;\,2\;\;3}\\ {3\;\;1\;\;1} \end{array}} \right]

View answer and explanation

Correct option: A

\left[ {\begin{array}{cccccccccccccccccccc} {\frac{1}{2}}&{ - \frac{1}{2}}&{\frac{1}{2}}\\ { - 4}&3&{ - 1}\\ {\frac{5}{2}}&{ - \frac{3}{2}}&{\frac{1}{2}} \end{array}} \right]

The correct answer is A. The inverse of a matrix is found by dividing the adjoint by the determinant. After calculating the determinant and adjoint, the inverse is found to be option A.

911 Easy

911 of 922

Analyzing +4 | -1

For the system of linear equations

\begin{array}{l} 2 x+4 y+2 a z=b \\ x+2 y+3 z=4 \\ 2 x-5 y+2 z=8\end{array}

which of the following is NOT correct?

View answer and explanation

Correct option: B

It has infinitely many solutions if

a=3, b=6

The correct answer is A (It has unique solution if

a=b=6

) because when a=3 and b=6, the system has no solution, not infinitely many.

912 Easy

912 of 922

Understanding +4 | -1

Let

A

be a skew-symmetric matrix of odd order, then

\left| A \right|

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the determinant of any skew-symmetric matrix of odd order is always zero.

913 Easy

913 of 922

Analyzing +4 | -1

a \ne b \ne c

, the value of

x

which satisfies the equation

\left| {\begin{array}{cccccccccccccccccccc} 0&{x - a}&{x - b}\\ {x + a}&0&{x - c}\\ {x + b}&{x + c}&0 \end{array}} \right| = 0,

View answer and explanation

Correct option: A

x = 0

Option A (

x = 0

) is correct because substituting

x = 0

into the determinant makes it equal to zero.

914 Easy

914 of 922

Application +4 | -1

P=\left[\begin{array}{cc} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{-1}{2} & \frac{\sqrt{3}}{2} \end{array}\right], A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right] \text { and } Q=P A P^{\mathrm{T}}

and

X=P^{\mathrm{T}} Q^{2010} P

, then

X

is equal to

View answer and explanation

Correct option: D

\left[\begin{array}{cc}1 & 2010 \\ 0 & 1\end{array}\right]

Option D (

\left[\begin{array}{cc}1 & 2010 \\ 0 & 1\end{array}\right]

) is correct because after simplifying the expression

X=P^{\mathrm{T}} Q^{2010} P

, it becomes

A^{2010}

, which equals

\left[\begin{array}{cc}1 & 2010 \\ 0 & 1\end{array}\right]

915 Easy

915 of 922

Analyzing +4 | -1

Let three matrices

A=\left[\begin{array}{ll}2 & 1 \\ 4 & 1\end{array}\right], B=\left[\begin{array}{ll}3 & 4 \\ 2 & 3\end{array}\right]

and

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

then

\operatorname{tr}(A)+\operatorname{tr}\left(\frac{A B C}{2}\right)+\operatorname{tr}\left(\frac{A(B C)^2}{4}\right)

+\operatorname{tr}\left(\frac{A(B C)^3}{8}\right)+\ldots+\infty

equals to

View answer and explanation

Correct option: D

Option D (6) is correct because after simplifying the expression using the property BC=I, the trace of the resulting geometric series converges to 6.

Explore Other Chapters in MATHEMATICS

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

916 Easy

916 of 922

Understanding +4 | -1

A

and

B

two are

3 \times 3

matrices, then which one of the following is not true:

View answer and explanation

Correct option: B

(A B)^{\prime}=A^{\prime} B^{\prime}

Option B (

(A B)^{\prime}=A^{\prime} B^{\prime}

) is not true because the transpose of a product of matrices is the product of their transposes in reverse order, i.e.,

(A B)^{\prime}=B^{\prime} A^{\prime}

917 Easy

917 of 922

Application +4 | -1

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

, then

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

, if the value of

\alpha

View answer and explanation

Correct option: B

\frac{\pi}{3}

Option B (

\frac{{\pi}}{{3}}

) is correct because when A + A' = I, it leads to

2\cos{\alpha} = 1

, which gives

\alpha = \frac{{\pi}}{{3}}

918 Easy

918 of 922

Application +4 | -1

\log _3 2+\log _3\left(2^x-7 / 2\right)=2 \log _3\left(2^x-5\right)

, then the value of

x

View answer and explanation

Correct option: A

Option A (3) is correct because it is the only value of x that satisfies the given logarithmic equation and the domain restrictions.

919 Easy

919 of 922

Application +4 | -1

A

is an involuntary matrix given by

A=\left[\begin{array}{rrr}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4\end{array}\right]

, then the inverse of

\frac{1}{2} A

will be

View answer and explanation

Correct option: A

2 A

Option A (2A) is correct because, for an involutory matrix A, A² = I, which implies A = A⁻¹. Therefore, the inverse of (1/2)A is 2A.

920 Easy

920 of 922

Analyzing +4 | -1

If for some

\alpha

and

\beta

R

, the intersection of the following three planes

\begin{array}{l} x + 4y - 2z = 1\\ x + 7y - 5z = \beta \\ x + 5y + az = 5 \end{array}

is a line in

{R^3}

, then

\alpha + \beta

is equal to

View answer and explanation

Correct option: B

Option B (10) is correct because it is the sum of alpha and beta after finding their values based on the condition that the intersection of the three planes is a line.

921 Easy

921 of 922

Analyzing +4 | -1

The rank of the matrix

\left[ {\begin{array}{cccccccccccccccccccc} 1\\ 2\\ 3\\ 6 \end{array}\;\,\,\begin{array}{cccccccccccccccccccc} 2\\ 4\\ 2\\ 8 \end{array}\;\,\,\begin{array}{cccccccccccccccccccc} 3\\ 3\\ 1\\ 7 \end{array}\;\,\,\begin{array}{cccccccccccccccccccc} 0\\ 2\\ 3\\ 5 \end{array}} \right]

View answer and explanation

Correct option: C

Option C (3) is correct because after row reduction, there are 3 non-zero rows, indicating a rank of 3.

922 Easy

922 of 922

Application +4 | -1

\left| {\begin{array}{cccccccccccccccccccc} \mathrm{\alpha }&{ - \mathrm{\beta }}&0\\ 0&\mathrm{\alpha }&\mathrm{\beta }\\ \mathrm{\beta }&0&\mathrm{\alpha } \end{array}} \right| = 0

, then

View answer and explanation

Correct option: A

\frac{\mathrm{\alpha }}{\mathrm{\beta }}

is one of the cube roots of unity

Option A (

\frac{{\rm{{\alpha }}}}{{\rm{{\beta }}}}}

is one of the cube roots of unity) is correct because the determinant equation simplifies to

(\frac{{\rm{{\alpha }}}}{{\rm{{\beta }}}})^3 = 1

, indicating that

\frac{{\rm{{\alpha }}}}{{\rm{{\beta }}}}

is a cube root of unity.

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