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

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

A and $B$ are two given matrices such that the order of $A$ is $3 \times 4$, if $\mathrm{A}^{\prime} \mathrm{B}$ and $\mathrm{BA}^{\prime}$ are both defined then

order of $B^{\prime} A$ is $4 \times 4$

order of $B^{\prime}$ is $3 \times 4$

order of $B^{\prime} A$ is $3 \times 3$

$B^{\prime} \mathrm{A}$ is undefined

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JEE • MATHEMATICS • MATRICES

Matrices practice · Easy

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571 Easy

571 of 922

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.

572 Easy

572 of 922

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.

573 Easy

573 of 922

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.

574 Easy

574 of 922

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.

575 Easy

575 of 922

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.

576 Easy

576 of 922

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.

577 Easy

577 of 922

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.

578 Easy

578 of 922

Analyzing +4 | -1

\mathrm{A}

and

\mathrm{B}

are symmetric matrices of the same order, then which one of the following is not true

View answer and explanation

Correct option: D

\mathrm{AB}-\mathrm{BA}

is symmetric

Option D ( 

\mathrm{AB}-\mathrm{BA}

is symmetric) is not true because the transpose of AB-BA is BA-AB, which is the negative of AB-BA, meaning it's skew-symmetric, not symmetric.

579 Easy

579 of 922

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.

580 Easy

580 of 922

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.

581 Easy

581 of 922

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.

582 Easy

582 of 922

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

583 Easy

583 of 922

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.

584 Easy

584 of 922

Analyzing +4 | -1

The rank of matrix

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

View answer and explanation

Correct option: C

Option C (2) is correct because the rank is determined by finding the largest non-zero minor of the matrix.

585 Easy

585 of 922

Analyzing +4 | -1

a,b,c

be respectively the

p\mathrm{th},q\mathrm{th}

and

r\mathrm{th}

terms of a HP., then

\Delta \; = \left| {\begin{array}{cccccccccccccccccccc} {bc}&{ca}&{ab}\\ p&q&r\\ 1&1&1 \end{array}} \right|

equals

View answer and explanation

Correct option: B

0

Option B (

0

) is correct because the determinant is zero, which is shown by expressing the reciprocals of the HP terms as an AP and manipulating the determinant.

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586 Easy

586 of 922

Application +4 | -1

A

is an involutory matrix given by

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

the inverse of

\frac{A}{2}

will be

View answer and explanation

Correct option: A

2 \mathrm{~A}

Option A (2A) is correct because for an involutory matrix A, A = A^-1, so (A/2)^-1 = 2A^-1 = 2A.

587 Easy

587 of 922

Understanding +4 | -1

For any square matrix

A,\;A{A^T}

is a

View answer and explanation

Correct option: B

Symmetric matrix

Option B (Symmetric matrix) is correct because the transpose of AA^T is equal to itself, satisfying the definition of a symmetric matrix.

588 Easy

588 of 922

Application +4 | -1

\Delta=\left|\begin{array}{ccc}-a & 2 b & 0 \\ 0 & -a & 2 b \\ 2 b & 0 & -a\end{array}\right|=0

, then

View answer and explanation

Correct option: D

a/b

is a cube root of 8

Option D (

a/b

is a cube root of 8) is correct because the determinant equals zero when (a/b)³ = 8, meaning a/b is a cube root of 8.

589 Easy

589 of 922

Application +4 | -1

If the matrix

{M_r}

is given by

{M_r} = \left[ {\begin{array}{cccccccccccccccccccc} r&{r - 1}\\ {r - 1}&r \end{array}} \right], = 1,2,3{,_.} \ldots ,

then the value of det

\left( {{M_1}} \right) + \det \left( {{M_2}} \right){ + _.} \ldots + \det \left( {{M_{2008}}} \right)

View answer and explanation

Correct option: C

{\left( {2008} \right)^2}

Option C (

(2008)^2

) is correct because the sum of the determinants forms an arithmetic progression, and its sum equals

(2008)^2

590 Easy

590 of 922

Understanding +4 | -1

If a matrix

A

is both symmetric and skew-symmetric, then

View answer and explanation

Correct option: C

A

is a zero matrix

Option C (

A

is a zero matrix) is correct because if a matrix is both symmetric and skew-symmetric, it must be the zero matrix.

591 Easy

591 of 922

Application +4 | -1

The rank of

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

View answer and explanation

Correct option: D

Option D is correct. The determinant of the matrix is non-zero, indicating that the rank of the matrix is 3.

592 Easy

592 of 922

Application +4 | -1

The values of

\lambda

for which the matrix

A=\left(\begin{array}{ccc}\lambda & 0 & \lambda \\ \lambda & 0 & -\lambda \\ 0 & 1 & 0\end{array}\right)

is orthogonal is

View answer and explanation

Correct option: D

\pm 1 / \sqrt{2}

Option D (

\pm 1 / \sqrt{2}

) is correct because for the matrix to be orthogonal, A*A' = I, which leads to the condition

\lambda = \pm 1 / \sqrt{2}

593 Easy

593 of 922

Application +4 | -1

The matrix

A = \left[ {\begin{array}{cccccccccccccccccccc} {\frac{1}{{\sqrt {2\;\;} }}\;\;\;\;\;\frac{2}{{\sqrt 2 }}}\\ { - \frac{1}{{\sqrt 2 }}\;\; - \frac{1}{{\sqrt 2 }}} \end{array}} \right]

View answer and explanation

Correct option: C

Nilpotent

Option C (Nilpotent) is correct because A² results in a zero matrix, satisfying the definition of a nilpotent matrix.

594 Easy

594 of 922

Analyzing +4 | -1

a>b>c

and the system of equations

a x+b y+c z=0, b x+c y+a z=0

and

c x+a y+b z=0

has a non-trivial solution, then the quadratic equation

ax^2+b x+c=0

has

View answer and explanation

Correct option: A

At least one positive root

Option A (At least one positive root) is correct because the condition a+b+c=0 implies that x=1 is a root of the quadratic equation ax²+bx+c=0, which is a positive root.

595 Easy

595 of 922

Application +4 | -1

X

is a

2 \times 3

matrix such that

\left[X^{\prime} X\right] \neq 0

, and

A

=I-X\left(X^{\prime} X\right)^{-1} X^{\prime}

then

A^2

is equal to

View answer and explanation

Correct option: A

A

Option A (A) is correct because by expanding A² and using the given expression for A, we find that A² = A.

596 Easy

596 of 922

Understanding +4 | -1

If for the matrix

\mathrm{A}, \mathrm{A}^3=\mathrm{I}

, then

\mathrm{A}^{-1}=

View answer and explanation

Correct option: A

\mathrm{A}^2

Option A (A²) is correct because if A³ = I, multiplying both sides by A^-1 gives A² = A^-1.

597 Easy

597 of 922

Understanding +4 | -1

A

and

B

are two matrices of the order

3 \times m

and

3 \times n

, respectivelyand

m=n

, then order of mamtrix

(5 A-2 B)

View answer and explanation

Correct option: D

3 \times n

Option D (3 x n) is correct because if m=n, both matrices A and B have the same order (3 x n), and the resulting matrix (5A - 2B) will also have the same order.

598 Easy

598 of 922

Analyzing +4 | -1

A and

B

are two given matrices such that the order of

A

3 \times 4

, if

\mathrm{A}^{\prime} \mathrm{B}

and

\mathrm{BA}^{\prime}

are both defined then

View answer and explanation

Correct option: B

order of

B^{\prime} A

4 \times 4

Option B is correct because the order of

B^{\prime}A

is determined by the compatibility of matrix multiplication, resulting in a

4 \times 4

matrix.

599 Easy

599 of 922

Understanding +4 | -1

A

2 \times 2

matrix such that

A^2=O

, then

\operatorname{tr}(A)

View answer and explanation

Correct option: C

Option C (0) is correct because if

A^2 = O

, then the eigenvalues of

A

are 0, and the trace of

A

is the sum of its eigenvalues.

600 Easy

600 of 922

Analyzing +4 | -1

Let

a-2 b+c=1

.
If

f(x)=\left|\begin{array}{ccc}x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3\end{array}\right|

, then:

View answer and explanation

Correct option: D

f(50)=1

Option D (

f(50)=1

) is correct because after applying row operations, the function simplifies to a constant value of 1.

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