Types of Matrices

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JEE Main 2011

LEVELBoard

Let $A$ and $B$ be two symmetric matrices of order 3. \\ Statement-1: $A (B A)$ and $(A B) A$ are symmetric matrices. \\ Statement-2: $A B$ is symmetric matrix if matrix multiplication of $A$ with $B$ is commutative.

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

JEE Main 2010

LEVELJEE Main

The number of $3 \times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$ , is

(A)

5

(B)

6

(C)

at least

7

(D)

less than

4

JEE Main 2004

LEVELBoard

Let $A = 00 - 1 0 - 1 0 - 1 00$ . The only correct statement about the matrix $A$ is

(A)

A^{2} = I

(B)

A = (- 1) I

where

I

is a unit matrix

(C)

A^{- 1}

does not exist

(D)

A

is a zero matrix

JEE Advanced 2015

LEVELBoard

Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$ , non zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

* Multiple Correct Options

(A)

Y^{3} Z^{4} - Z^{4} Y^{3}

(B)

X^{44} + Y^{44}

(C)

X^{4} Z^{3} - Z^{3} X^{4}

(D)

X^{23} + Y^{23}

JEE Advanced 2010

LEVELJEE Advanced

Comprehension Passage

Let

p

be an odd prime number and

T_{p}

be the following set of

2 \times 2

matrices :

T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}

Question 1:

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

(A)

(p - 1)^{2}

(B)

2 (p - 1)

(C)

(p - 1)^{2} + 1

(D)

2 p - 1

Question 2:

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

(A)

(p - 1) (p^{2} - p + 1)

(B)

p^{3} - (p - 1)^{2}

(C)

(p - 1)^{2}

(D)

(p - 1) (p - 2)

Question 3:

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

(A)

2 p^{2}

(B)

p^{3} - 5 p

(C)

p^{3} - 3 p

(D)

p^{3} - p^{2}

JEE Main 2019 (11 January)

LEVELBoard

Let $A = 0 p p 2 q q - q r - r r$ . If $A A^{T} = I_{3}$ , then $∣ p ∣$ is :

(A)

\frac{1}{2}

(B)

\frac{1}{5}

(C)

\frac{1}{6}

(D)

\frac{1}{3}

JEE Main 2019 (12 April Shift 1)

LEVELBoard

If A is a symmetric matrix and B is a skew-symmetric matrix such that $A + B = (25 3 - 1)$ , then AB is equal to :

(A)

(- 4 1 24)

(B)

(- 4 - 1 - 2 4)

(C)

(4 - 1 - 2 - 4)

(D)

(41 2 - 4)

JEE Main 2020 - 2 Sep (Morning)

LEVELJEE Main

Let $A$ be a $2 \times 2$ real matrix with entries from ${0, 1}$ and $∣ A ∣ \neq = 0$ . Consider the following two statements : (P) If $A \neq = I_{2}$ , then $∣ A ∣ = - 1$ (Q) If $∣ A ∣ = 1$ , then $t r (A) = 2$ , where $I_{2}$ denotes $2 \times 2$ identity matrix and $t r (A)$ denotes the sum of the diagonal entries of $A$ . Then:

(A)

(P) is true and (Q) is false

(B)

Both (P) and (Q) are false

(C)

Both (P) and (Q) are true

(D)

(P) is false and (Q) is true

JEE Main 2020 - 2 Sep (Evening)

LEVELJEE Main

Let $a, b, c \in R$ be all non-zero and satisfy $a^{3} + b^{3} + c^{3} = 2$ . If the matrix $A = a b c b c a c a b$ satisfies $A A^{T} = I$ , then a value of $ab c$ can be :

(A)

3

(B)

\frac{1}{3}

(C)

- \frac{1}{3}

(D)

\frac{2}{3}

JEE Main 2021 (31 Aug Shift 2)

LEVELJEE Main

The number of elements in the set ${A = (a 0 b d) : a, b, d \in {- 1, 0, 1} and (I - A)^{3} = I - A^{3}}$ , where $I$ is $2 \times 2$ identity matrix, is :

JEE Main 2021 (20 July Shift 1)

LEVELJEE Main

Let $A = [2 a 30], a \in R$ be written as $P + Q$ where $P$ is a symmetric matrix and $Q$ is a skew symmetric matrix. If $det (Q) = 9$ , then the modulus of the sum of all possible values of determinant of $P$ is equal to :

JEE Main 2021 (26 February Shift 1)

LEVELJEE Main

Let $A$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^{2}$ is 1, then the possible number of such matrices is:

JEE Main 2021 (16 March Shift 1)

LEVELJEE Main

The total number of $3 \times 3$ matrices $A$ having entries from the set ${0, 1, 2, 3}$ such that the sum of all the diagonal entries of $A A^{T}$ is 9, is equal to ______

JEE Main 2022 (26 July Shift 2)

LEVELJEE Main

The number of matrices $A = [a c b d]$ , where $a, b, c, d \in {- 1, 0, 1, 2, 3, \dots, 10}$ , such that $A = A^{- 1}$ , is

JEE Main 2022 (28 July Shift 2)

LEVELBoard

Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

(A)

A^{4} - B^{4}

is a symmetric matrix

(B)

A B - B A

is a symmetric matrix

(C)

B^{5} - A^{5}

is a skew-symmetric matrix

(D)

A B + B A

is a skew-symmetric matrix

JEE Main 2023 (01 February Shift 2)

LEVELJEE Main

If $A = \frac{1}{2} [1 - 3 31]$ , then :

(A)

A^{30} - A^{25} = 2 I

(B)

A^{30} + A^{25} + A = I

(C)

A^{30} + A^{25} - A = I

(D)

A^{30} = A^{25}

JEE Main 2023 (13 April Shift 1)

LEVELBoard

The number of symmetric matrices of order 3, with all the entries from the set ${0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$ is

(A)

6^{10}

(B)

1 0^{6}

(C)

9^{10}

(D)

1 0^{9}

JEE Main 2023 (24 January Shift 2)

LEVELJEE Main

The number of square matrices of order 5 with entries from the set ${0, 1}$ , such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is

JEE Main 2023 (25 January Shift 2)

LEVELJEE Main

Let $A, B, C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements (S1) $A^{13} B^{26} - B^{26} A^{13}$ is symmetric (S2) $A^{26} C^{13} - C^{13} A^{26}$ is symmetric. Then,

Both S1 and S2 are false

(D)

Both S1 and S2 are true

JEE Main 2023 (29 January Shift 2)

LEVELJEE Main

Let $A$ be a symmetric matrix such that $∣ A ∣ = 2$ and $[23 1 \frac{3}{2}] A = [1 α 2 β]$ . If the sum of the diagonal elements of $A$ is $s$ , then $\frac{β s}{α ^{2}}$ is equal to ______.

JEE Main 2024 (01 Feb Shift 2)

LEVELJEE Main

Let $A = I_{2} - 2 M M^{T}$ , where $M$ is real matrix of order $2 \times 1$ such that the relation $M^{T} M = I_{1}$ holds. If $λ$ is a real number such that the relation $A X = λ X$ holds for some non-zero real matrix $X$ of order $2 \times 1$ , then the sum of squares of all possible values of $λ$ is equal to :

JEE Main 2024 (29 Jan Shift 1)

LEVELBoard

Let $A$ be a square matrix such that $A A^{T} = I$ . Then $\frac{1}{2} A [(A + A^{T})^{2} + (A - A^{T})^{2}]$ is equal to

(A)

A^{2} + I

(B)

A^{3} + I

(C)

A^{2} + A^{T}

(D)

A^{3} + A^{T}

JEE Main 2025 April

LEVELJEE Main

The number of singular matrices of order 2, whose elements are from the set ${2, 3, 6, 9}$ is _____

JEE Main 2025 April

LEVELJEE Main

Let $A = [α 6 - 1 β]$ , $α > 0$ , such that $det (A) = 0$ and $α + β = 1$ . If $I$ denotes $2 \times 2$ identity matrix, then the matrix $(I + A)^{8}$ is:

(A)

[46 - 1 - 1]

(B)

[257514 - 64 - 127]

(C)

[10252024 - 511 - 1024]

(D)

[7661530 - 255 - 509]

JEE Main 2025 (January)

LEVELJEE Advanced

Let M denote the set of all real matrices of order $3 \times 3$ and let $S = {- 3, - 2, - 1, 1, 2}$ . Let $S_{1} = {A = [a_{ij}] \in M : A = A^{T} and a_{ij} \in S, \forall i, j}$ $S_{2} = {A = [a_{ij}] \in M : A = - A^{T} and a_{ij} \in S, \forall i, j}$ $S_{3} = {A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 and a_{ij} \in S, \forall i, j}$ If $n (S_{1} \cup S_{2} \cup S_{3}) = 125 α$ then $α$ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of 3×3 non-singular matrices, with four entries as 1 and all other entries as 0, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​00−1​0−10​−100​​. The only correct statement about the matrix A is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that the trace of A is not divisible by p but det(A) is divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that det(A) is not divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​0pp​2qq−q​r−rr​​. If AAT=I3​, then ∣p∣ is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A is a symmetric matrix and B is a skew-symmetric matrix such that A+B=(25​3−1​), then AB is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c∈R be all non-zero and satisfy a3+b3+c3=2. If the matrix A=​abc​bca​cab​​ satisfies AAT=I, then a value of abc can be :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of elements in the set {A=(a0​bd​):a,b,d∈{−1,0,1} and (I−A)3=I−A3}, where I is 2×2 identity matrix, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The total number of 3×3 matrices A having entries from the set {0,1,2,3} such that the sum of all the diagonal entries of AAT is 9, is equal to ______

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A=[ac​bd​], where a,b,c,d∈{−1,0,1,2,3,…,10}, such that A=A−1, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A and B be any two 3×3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=21​[1−3​​3​1​], then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of symmetric matrices of order 3, with all the entries from the set {0,1,2,3,4,5,6,7,8,9} is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a symmetric matrix such that ∣A∣=2 and [23​123​​]A=[1α​2β​]. If the sum of the diagonal elements of A is s, then α2βs​ is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a square matrix such that AAT=I. Then 21​A[(A+AT)2+(A−AT)2] is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of singular matrices of order 2, whose elements are from the set {2,3,6,9} is _____

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=[α6​−1β​], α>0, such that det(A)=0 and α+β=1. If I denotes 2×2 identity matrix, then the matrix (I+A)8 is:

The number of $3 \times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$ , is

Let $A = 00 - 1 0 - 1 0 - 1 00$ . The only correct statement about the matrix $A$ is

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

Let $A = 0 p p 2 q q - q r - r r$ . If $A A^{T} = I_{3}$ , then $∣ p ∣$ is :

If A is a symmetric matrix and B is a skew-symmetric matrix such that $A + B = (25 3 - 1)$ , then AB is equal to :

Let $a, b, c \in R$ be all non-zero and satisfy $a^{3} + b^{3} + c^{3} = 2$ . If the matrix $A = a b c b c a c a b$ satisfies $A A^{T} = I$ , then a value of $ab c$ can be :

The number of elements in the set ${A = (a 0 b d) : a, b, d \in {- 1, 0, 1} and (I - A)^{3} = I - A^{3}}$ , where $I$ is $2 \times 2$ identity matrix, is :

Let $A$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^{2}$ is 1, then the possible number of such matrices is:

The total number of $3 \times 3$ matrices $A$ having entries from the set ${0, 1, 2, 3}$ such that the sum of all the diagonal entries of $A A^{T}$ is 9, is equal to ______

The number of matrices $A = [a c b d]$ , where $a, b, c, d \in {- 1, 0, 1, 2, 3, \dots, 10}$ , such that $A = A^{- 1}$ , is

Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

If $A = \frac{1}{2} [1 - 3 31]$ , then :

The number of symmetric matrices of order 3, with all the entries from the set ${0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$ is

Let $A$ be a symmetric matrix such that $∣ A ∣ = 2$ and $[23 1 \frac{3}{2}] A = [1 α 2 β]$ . If the sum of the diagonal elements of $A$ is $s$ , then $\frac{β s}{α ^{2}}$ is equal to ______.

Let $A$ be a square matrix such that $A A^{T} = I$ . Then $\frac{1}{2} A [(A + A^{T})^{2} + (A - A^{T})^{2}]$ is equal to

The number of singular matrices of order 2, whose elements are from the set ${2, 3, 6, 9}$ is _____

Let $A = [α 6 - 1 β]$ , $α > 0$ , such that $det (A) = 0$ and $α + β = 1$ . If $I$ denotes $2 \times 2$ identity matrix, then the matrix $(I + A)^{8}$ is: