If ${\Delta _1} = \left| {\begin{array}{*{20}{c}} x&{\sin \theta }&{\cos \theta }\\ { - \sin \theta }&{ - x}&1\\ {\cos \theta }&1&x \end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}} x&{\sin 2\theta }&{\cos 2\theta }\\ { - \sin 2\theta }&{ - x}&1\\ {\cos 2\theta }&1&x \end{array}} \right|,\,\,x \ne 0,$ then for all $\theta \in \left( {0,\frac{\pi }{2}} \right)$

${\Delta _1} + {\Delta _2} = - 2\left( {{x^3} + x - 1} \right)$

${\Delta _1} - {\Delta _2} = x\left( {\cos 2\theta - \cos 4\theta } \right)$

If $F\left( x \right) = \left[ {\begin{array}{*{20}{c}} {\cos x}&{ - \sin x}&0\\ {\sin x}&{\cos x}&0\\ 0&0&1 \end{array}} \right]\;{\rm{and\;}}G\left( y \right) = \left[ {\begin{array}{*{20}{c}} {\cos x}&0&{\sin x}\\ 0&1&0\\ { - \sin x}&0&{\cos x} \end{array}} \right],$ then ${\left[ {F\left( x \right)G\left( y \right)} \right]^{ - 1}}$ is equal to

$F\left( {{x^{ - 1}}} \right)G\left( {{y^{ - 1}}} \right)$

$G\left( {{y^{ - 1}}} \right)F\left( {{x^{ - 1}}} \right)$

If A is a symmetric matrix and B is a skew – symmetric matrix such that $A + B = \left[ {\begin{array}{*{20}{c}} 2&3\\ 5&{ - 1} \end{array}} \right]$, then AB is equal to

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

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

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

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

The value of the determinant $\left| {\begin{array}{*{20}{c}} 1&1&1\\ 1&{1 - x}&1\\ 1&1&{1 + y} \end{array}} \right|$ is equal to

$\left( {1 - x} \right)\left( {1 + y} \right)$

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JEE • MATHEMATICS • MATRICES

Matrices practice · Mixed difficulty

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601 Medium

601 of 995

Understanding +4 | -1

A

be a square matrix of order

n

and if

\left| A \right| = D

and

\left| {\mathrm{adj\;A}} \right| = D'

, then

View answer and explanation

Correct option: C

DD' = {D^n}

Option C (

DD' = {D^n}

) is correct because |adj A| = |A|^(n-1) and |A| = D, |adj A| = D', so D' = D^(n-1). Therefore, DD' = D * D^(n-1) = D^n.

602 Easy

602 of 995

Application +4 | -1

A=\left[\begin{array}{cc}1 & 2 \\ 2 & 1\end{array}\right]

and

f(x)=\frac{1+x}{1-x}

, then

f(A)

View answer and explanation

Correct option: C

\left[\begin{array}{ll}-1 & -1 \\ -1 & -1\end{array}\right]

Option C (

\left[\begin{array}{ll}-1 & -1 \\ -1 & -1\end{array}\right]

) is correct because by substituting the matrix A into the function f(x) and simplifying, we arrive at this result.

603 Easy

603 of 995

Analyzing +4 | -1

{\Delta _1} = \left| {\begin{array}{cccccccccccccccccccc} x&{\sin \theta }&{\cos \theta }\\ { - \sin \theta }&{ - x}&1\\ {\cos \theta }&1&x \end{array}} \right|

and

{\Delta _2} = \left| {\begin{array}{cccccccccccccccccccc} x&{\sin 2\theta }&{\cos 2\theta }\\ { - \sin 2\theta }&{ - x}&1\\ {\cos 2\theta }&1&x \end{array}} \right|,\,\,x \ne 0,

then for all

\theta \in \left( {0,\frac{\pi }{2}} \right)

View answer and explanation

Correct option: C

{\Delta _1} + {\Delta _2} = - 2{x^3}

Option C (

\Delta _1 + \Delta _2 = - 2{x^3}

) is correct because the sum of the determinants simplifies to -2x³ after expansion.

604 Easy

604 of 995

Understanding +4 | -1

A

is a non-singular matrix of order 3 , then

|\operatorname{adj} A|=|A|^{\mathrm{n}}

. Here the value of

n

View answer and explanation

Correct option: A

Option A (2) is correct because for a non-singular matrix A of order 3, |adj A| = |A|^(3-1) = |A|^2.

605 Easy

605 of 995

Understanding +4 | -1

A

is a singular matrix, then

A

adj (A) is a

View answer and explanation

Correct option: B

Zero matrix

The correct answer is B (Zero matrix) because for a singular matrix A, its determinant is 0, and A adj(A) equals |A|I, which results in a zero matrix.

606 Medium

606 of 995

Application +4 | -1

\left| {\begin{array}{cccccccccccccccccccc} 1&2&3\\ {{1^3}}&{{2^3}}&{{3^3}}\\ {{1^5}}&{{2^5}}&{{3^5}} \end{array}} \right|

is equal to

View answer and explanation

Correct option: C

6!

Option C (

6!

) is correct because the determinant simplifies to 720, which is equal to 6 factorial.

607 Easy

607 of 995

Analyzing +4 | -1

a + x = b + y = c + z + 1,\,

where

a,\,b,\,c,\,x,\,y,\,z

are non-zero distinct real numbers, then

\left| {\begin{array}{cccccccccccccccccccc} x&{a + y}&{x + a}\\ y&{b + y}&{y + b}\\ z&{c + y}&{z + c} \end{array}} \right|

is equal to

View answer and explanation

Correct option: B

y(a - b)

Option B (

y(a - b)

) is correct because the determinant can be simplified using properties of determinants and the given conditions to arrive at this expression.

608 Easy

608 of 995

Application +4 | -1

If a square matrix

A

is such that

A{A^T} = I = {A^T}A\,\,\,\mathrm{then}\,\,\,\,\left| A \right|

then is equal to

View answer and explanation

Correct option: B

\pm \,1

Option B (

\pm 1

) is correct because the determinant of a matrix multiplied by its transpose equals 1, implying the determinant of the matrix is either 1 or -1.

609 Easy

609 of 995

Application +4 | -1

\mathrm{A}=\left[\begin{array}{cc}0 & 2 \\ 3 & -4\end{array}\right]

and

k \mathrm{~A}=\left[\begin{array}{cc}0 & 3 a \\ 2 b & 24\end{array}\right]

, then the values of

k, a, b

are respectively

View answer and explanation

Correct option: C

-6,-4,-9

Option B (-6, -4, -9) is correct because equating the elements of kA with the given matrix allows us to solve for k, a, and b.

610 Easy

610 of 995

Application +4 | -1

\left[ {1\;\;x\;\;1} \right]\left[ {\begin{array}{cccccccccccccccccccc} 1&2&3\\ 0&5&1\\ 0&3&2 \end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc} x\\ 1\\ { - 2} \end{array}} \right] = 0

, then the value of

x

View answer and explanation

Correct option: C

\frac{5}{4}

Option C (

\frac{5}{4}

) is correct because solving the matrix equation for x results in x = 5/4.

611 Medium

611 of 995

Analyzing +4 | -1

D_r=\left|\begin{array}{lll}r & 1 & \frac{n(n+1)}{2} \\ 2 r-1 & 4 & n^2 \\ 2^{r-1} & 5 & 2^n-1\end{array}\right|

, then the value of

\sum\limits_{r = 1}^n {{D_r},}

View answer and explanation

Correct option: A

Option A (0) is correct because the summation of the determinants results in a determinant with two identical columns, making its value zero.

612 Medium

612 of 995

Application +4 | -1

The value of

\left| {\begin{array}{cccccccccccccccccccc} {(a + 1)\left( {a + 2} \right)}&{a + 2}&1\\ {\left( {a + 2} \right)\left( {a + 3} \right)}&{a + 3}&1\\ {\left( {a + 3} \right)\left( {a + 4} \right)}&{a + 4}&1 \end{array}} \right|

View answer and explanation

Correct option: B

- 2

Option B (-2) is correct because after applying row operations and expanding the determinant, the expression simplifies to -2.

613 Easy

613 of 995

Application +4 | -1

\left| {\begin{array}{cccccccccccccccccccc} { - 12}&0&\lambda \\ 0&2&{ - 1}\\ 2&1&{15} \end{array}} \right| = - 360

, then the value of

\lambda

View answer and explanation

Correct option: C

- 3

Option C (-3) is correct because solving the determinant equation for

\lambda

yields -3.

614 Easy

614 of 995

Understanding +4 | -1

The number of all possible matrices of order

3 \times 3

with each entry 0 or 1 is

View answer and explanation

Correct option: D

512

Option D (512) is correct because each of the 9 entries in the 3x3 matrix can be either 0 or 1, leading to 2⁹ possible matrices.

615 Medium

615 of 995

Application +4 | -1

The roots of the equation

\left| {\begin{array}{cccccccccccccccccccc} 1&4&{20}\\ 1&{ - 2}&5\\ 1&{2x}&{5{x^2}} \end{array}} \right| = 0

are

View answer and explanation

Correct option: B

- 1,2

Option B (-1, 2) is correct because these are the roots obtained after simplifying the determinant equation and solving for x.

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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

616 Medium

616 of 995

Application +4 | -1

F\left( x \right) = \left[ {\begin{array}{cccccccccccccccccccc} {\cos x}&{ - \sin x}&0\\ {\sin x}&{\cos x}&0\\ 0&0&1 \end{array}} \right]\;\mathrm{and\;}G\left( y \right) = \left[ {\begin{array}{cccccccccccccccccccc} {\cos x}&0&{\sin x}\\ 0&1&0\\ { - \sin x}&0&{\cos x} \end{array}} \right],

then

{\left[ {F\left( x \right)G\left( y \right)} \right]^{ - 1}}

is equal to

View answer and explanation

Correct option: C

G\left( { - y} \right)F\left( { - x} \right)

Option C (

G\left( { - y} \right)F\left( { - x} \right)

) is correct because the inverse of the product of two matrices is the product of their inverses in reverse order.

617 Medium

617 of 995

Understanding +4 | -1

A=\left[a_{i j}\right]

is a matrix of order

4 \times 5

, then the diagonal elements of

\mathrm{A}

are

View answer and explanation

Correct option: D

do not exist

Option D (do not exist) is correct because a diagonal requires a square matrix, and a 4x5 matrix is not square.

618 Easy

618 of 995

Application +4 | -1

Let

A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5\end{array}\right]

. If

\left|A^2\right|=25

then

|\alpha|

equals

View answer and explanation

Correct option: C

1 / 5

Option C (

\frac{1}{5}

) is correct because the determinant of A squared equals 25, leading to the absolute value of alpha being 1/5.

619 Medium

619 of 995

Application +4 | -1

The values of

x

for which the given matrix

\left[ {\begin{array}{cccccccccccccccccccc} { - x}&x&2\\ 2&x&{ - x}\\ x&{ - 2}&{ - x} \end{array}} \right]

will be non-singular, are

View answer and explanation

Correct option: B

For all

x

other then 2 and

- 2

Option B (For all x other then 2 and -2) is correct because the matrix is non-singular when its determinant is non-zero, which occurs for all x except 2 and -2.

620 Medium

620 of 995

Analyzing +4 | -1

Consider the system of equations

\alpha x+y+z=p, x+\alpha y+z=q

and

x+y+\alpha z=r

, then the sum of all possible distinct value(s) of

\alpha

for which system does not possess a unique solution is

View answer and explanation

Correct option: C

-1

Option C (-1) is correct because the sum of all possible distinct values of

\alpha

for which the system does not possess a unique solution is -1, derived from the determinant of the coefficient matrix.

621 Easy

621 of 995

Application +4 | -1

A = \left[ {\begin{array}{cccccccccccccccccccc} {ab\;\;\;\;\;\;\;{b^2}}\\ { - {a^2}\;\; - ab} \end{array}} \right],

then

A

is equal to

View answer and explanation

Correct option: C

Nilpotent

Option C (Nilpotent) is correct because when matrix A is multiplied by itself, the result is a zero matrix.

622 Easy

622 of 995

Application +4 | -1

The value of the determinant

\left| {\begin{array}{cccccccccccccccccccc} 1&{{\omega ^3}}&{{\omega ^5}}\\ {{\omega ^3}}&1&{{\omega ^4}}\\ {{\omega ^5}}&{{\omega ^4}}&1 \end{array}} \right|,

where

\omega

is an imaginary cube root of unity, is

View answer and explanation

Correct option: B

Option B (3) is correct because the determinant simplifies to 3 using the properties of complex cube roots of unity.

623 Easy

623 of 995

Application +4 | -1

A=\left|\begin{array}{ccc}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right|

, then

A^{-1}

exists, if

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the condition for A inverse to exist is

\lambda \neq -8/5

, which is not covered by options A, B, or C.

624 Medium

624 of 995

Application +4 | -1

Let

A

be a

3 \times 3

matrix such that

|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))|

=12^{4}

. Then

\mid A^{-1} \, \operatorname{adj} \, A \mid

is equal to

View answer and explanation

Correct option: B

2 \sqrt{3}

Option B (

2 \sqrt{3}

) is correct because it is derived from the properties of adjoint and determinant of a matrix.

625 Easy

625 of 995

Application +4 | -1

If A is a symmetric matrix and B is a skew – symmetric matrix such that

A + B = \left[ {\begin{array}{cccccccccccccccccccc} 2&3\\ 5&{ - 1} \end{array}} \right]

, then AB is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of correctly finding matrices A and B and then multiplying them to find AB.

626 Easy

626 of 995

Application +4 | -1

Given,

f(x)=\log _{10} x

and

g(x)=e^{\pi i x}

. If

\phi(x)=\left|\begin{array}{ccc}f(x) \cdot g(x) & (f(x))^{g(x)} & 1 \\ f\left(x^2\right) \cdot g\left(x^2\right) & \left(f\left(x^2\right)\right)^{g\left(x^2\right)} & 0 \\ f\left(x^3\right) \cdot g\left(x^3\right) & \left(f\left(x^3\right)\right)^{g\left(x^3\right)} & 1\end{array}\right|

, the value of

\phi(10)

, is

View answer and explanation

Correct option: C

Option C (0) is correct because the determinant evaluates to 0 after substituting the given functions and values.

627 Easy

627 of 995

Analyzing +4 | -1

Let

a, b, c

be any real numbers. Suppose that there are real numbers

x, y, z

not all zero such that

x=c y+b z

y=a z+c x

and

z=b x+a y

. Then,

a^2+b^2+c^2+2 a b c

is equal to

View answer and explanation

Correct option: D

Option D is correct because the given condition implies that the determinant of the coefficient matrix must be zero, leading to

a^2 + b^2 + c^2 + 2abc = 1

628 Hard

628 of 995

Analyzing +4 | -1

The number of positive integral solution of the equation

\left|\begin{array}{ccc}x^3+1 & x^2 y & x^2 z \\ x y^2 & y^3+1 & y^2 z \\ x z^2 & y z^2 & z^3+1\end{array}\right|=11

View answer and explanation

Correct option: B

The correct answer is B (3) because solving the determinant equation leads to x³ + y³ + z³ = 10, which has three positive integral solutions.

629 Medium

629 of 995

Application +4 | -1

a > 0,b > 0,c > 0

are respectively the

{p^{th}},{q^{th}},{r^{th}}

terms of a GP, then the value of the determinant

\left| {\begin{array}{cccccccccccccccccccc} {\log a}&p&1\\ {\log b}&q&1\\ {\log c}&r&1 \end{array}} \right|,

View answer and explanation

Correct option: B

Option B (0) is correct because expressing log a, log b, and log c in terms of log A and log R and applying column operations results in a zero determinant.

630 Easy

630 of 995

Application +4 | -1

The value of the determinant

\left| {\begin{array}{cccccccccccccccccccc} 1&1&1\\ 1&{1 - x}&1\\ 1&1&{1 + y} \end{array}} \right|

is equal to

View answer and explanation

Correct option: D

- xy

Option D (

-xy

) is correct because the determinant simplifies to

-xy

after applying column operations.

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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