Algebraic Operations on Matrices

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JEE Main 2015

If $A = 12 a 212 2 - 2 b$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to:

(2, 1)

(B)

(- 2, - 1)

(C)

(2, - 1)

(D)

(- 2, 1)

JEE Main 2014

LEVELJEE Main

If $A$ is a $3 \times 3$ non-singular matrix such that $A A^{'} = A^{'} A$ and $B = A^{- 1} A^{'}$ , then $B B^{'}$ equals:

(A)

B^{- 1}

(B)

(B^{- 1})^{'}

(C)

I + B

(D)

I

JEE Main 2012

LEVELBoard

Let $A = 123012001$ . If $u_{1}$ and $u_{2}$ are column matrices such that $A u_{1} = 100$ and $A u_{2} = 010$ , then $u_{1} + u_{2}$ is equal to:

(A)

- 1 10

(B)

- 1 1 - 1

(C)

- 1 - 1 0

(D)

1 - 1 - 1

JEE Main 2010

LEVELJEE Main

Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^{2} = I$ , where $I$ is $2 \times 2$ identity matrix. Define $T r (A) =$ sum of diagonal elements of $A$ and $∣ A ∣ =$ determinant of matrix $A$ . \\ Statement-1: $T r (A) = 0$ . \\ Statement-2: $∣ A ∣ = 1$ .

(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 2008

LEVELJEE Main

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $t r (A)$ , the sum of diagonal entries of $A$ . Assume that $A^{2} = I$ . \\ Statement-1: If $A \neq = I$ and $A \neq = - I$ , then $d e t (A) = - 1$ \\ Statement-2: If $A \neq = I$ and $A \neq = - I$ , then $t r (A) \neq = 0$ .

(A)

Statement-1 is false, Statement-2 is true

(B)

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

(C)

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

(D)

Statement-1 is true, Statement-2 is false

JEE Main 2006

LEVELBoard

If $A$ and $B$ are square matrices of size $n \times n$ such that $A^{2} - B^{2} = (A - B) (A + B)$ , then which of the following will be always true?

(A)

A = B

(B)

A B = B A

(C)

either of

A

B

is a zero matrix

(D)

either of

A

B

is identity matrix

JEE Main 2006

LEVELBoard

Let $A = [1324]$ and $B = [a 0 0 b]$ , $a, b \in N$ . Then

(A)

there cannot exist any

B

such that

A B = B A

(B)

there exist more than one but finite number of

B

's such that

A B = B A

(C)

there exists exactly one

B

such that

A B = B A

(D)

there exist infinitely many

B

's such that

A B = B A

JEE Main 2005

LEVELBoard

If $A = [1101]$ and $I = [1001]$ , then which one of the following holds for all $n \geq 1$ , by the principle of mathematical induction

(A)

A^{n} = n A - (n - 1) I

(B)

A^{n} = 2^{n - 1} A - (n - 1) I

(C)

A^{n} = n A + (n - 1) I

(D)

A^{n} = 2^{n - 1} A + (n - 1) I

JEE Main 2003

LEVELBoard

If $A = [a b b a]$ and $A^{2} = [α β β α]$ , then

(A)

α = 2 ab, β = a^{2} + b^{2}

(B)

α = a^{2} + b^{2}, β = ab

(C)

α = a^{2} + b^{2}, β = 2 ab

(D)

α = a^{2} + b^{2}, β = a^{2} - b^{2}

JEE Advanced 2016

LEVELJEE Main

Let $P = 1416014001$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$ , then $\frac{q _{31} + q _{32}}{q _{21}}$ equals

Let $ω$ be a complex cube root of unity with $ω \neq = 1$ and $P = [p_{ij}]$ be an $n \times n$ matrix with $p_{ij} = ω^{i + j}$ . Then $P^{2} \neq = 0$ , when $n =$

* Multiple Correct Options

(A)

57

(B)

55

(C)

58

(D)

56

JEE Advanced 2012

LEVELBoard

If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2 P + I$ , where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = x y z \neq = 000$ such that

(A)

(a)

P X = 000

(B)

(b)

P X = X

(C)

(c)

P X = 2 X

(D)

(d)

P X = - X

JEE Advanced 2011

LEVELJEE Main

Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N = N M$ . If $P^{T}$ denotes the transpose of $P$ , then $M^{2} N^{2} (M^{T} N)^{- 1} (M N^{- 1})^{T}$ is equal to

(A)

M^{2}

(B)

- N^{2}

(C)

- M^{2}

(D)

M N

JEE Advanced 2005

LEVELJEE Main

If $P = [\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2}]$ and $A = [1011]$ and $Q = P A P^{T}$ and $x = P^{T} Q^{2005} P$ then $x$ is equal to

(A)

(a)

[1020051]

(B)

(b)

[4 + 20053 2005 6015 4 - 20053]

(C)

(c)

\frac{1}{4} [2 + 3 - 1 1 2 - 3]

(D)

(d)

\frac{1}{4} [2005 2 + 3 2 - 3 2005]

JEE Advanced 2003

LEVELBoard

If $A = [α 1 01]$ and $B = [1501]$ , then value of $α$ for which $A^{2} = B$ , is

- 1

If $A$ and $B$ are square matrices of equal degree, then which one is correct among the followings?

(A)

(a)

A + B = B + A

(B)

(b)

A + B = A - B

(C)

(c)

A - B = B - A

(D)

(d)

A B = B A

JEE Advanced 2016

LEVELJEE Main

Let $z = \frac{- 1 + 3 i}{2}$ , where $i = - 1$ , and $r, s \in {1, 2, 3}$ . Let $P = [(- z)^{r} z^{2 s} z^{2 s} z^{r}]$ and I be the identity matrix of order 2. Then the total number of ordered pairs $(r, s)$ for which $P^{2} = - I$ is

JEE Advanced 2011

LEVELJEE Main

Let M be a $3 \times 3$ matrix satisfying $M 010 = - 1 23, M 1 - 1 0 = 11 - 1$ , and $M 111 = 0012$ . Then the sum of the diagonal entries of M is

JEE Advanced 2008

LEVELJEE Advanced

Match the Statements/Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

The minimum value of

\frac{x ^{2} + 2 x + 4}{x + 2}

(Q)

Let A and B be

3 \times 3

matrices of real numbers, where A is symmetric, B is skew-symmetric, and

(A + B) (A - B) = (A - B) (A + B)

. If

(A B)^{t} = (- 1)^{k} A B

, where

(A B)^{t}

is the transpose of the matrix AB, then the possible values of

k

are

(R)

Let

a = lo g_{3} lo g_{3} 2

. An integer

k

satisfying

1 < 2^{- k + 3^{- a}} < 2

, must be less than

(S)

sin θ = cos ϕ

, then the possible values of

\frac{1}{π} (θ \pm ϕ - \frac{π}{2})

are

List-II

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

Let A be a matrix such that $A \cdot [1023]$ is a scalar matrix and $∣3 A ∣ = 108$ . Then $A^{2}$ equals :

(A)

[360 - 32 4]

(B)

[4 - 32 036]

(C)

[40 - 32 36]

(D)

[36 - 32 04]

JEE Main 2018 (15 April Evening)

LEVELBoard

Suppose A is any $3 \times 3$ non-singular matrix and $(A - 3 I) (A - 5 I) = 0$ , where $I = I_{3}$ and $O = O_{3}$ . If $α A + β A^{- 1} = 4 I$ , then $α + β$ is equal to :-

JEE Main 2018 (16 April Shift 1)

LEVELBoard

Let $A = 111011001$ and $B = A^{20}$ . Then the sum of the elements of the first column of B is :

JEE Main 2019 (9 January)

LEVELJEE Main

If $A = [cos θ sin θ - sin θ cos θ]$ , then the matrix $A^{- 50}$ when $θ = \frac{π}{12}$ , is equal to :

(A)

[\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2}]

(B)

[\frac{1}{2} \frac{3}{2} \frac{3}{2} \frac{1}{2}]

(C)

[\frac{1}{2} \frac{3}{2} \frac{3}{2} \frac{1}{2}]

(D)

[\frac{1}{2} - \frac{3}{2} \frac{3}{2} \frac{1}{2}]

JEE Main 2019 (12 January Shift 1)

LEVELJEE Main

Let $P = 139013001$ and $Q = [q_{ij}]$ be two $3 \times 3$ matrices such that $Q - P^{5} = I_{3}$ . Then $\frac{q _{21} + q _{31}}{q _{32}}$ is equal to:

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

Let $A = [cos α sin α - sin α cos α], (α \in R)$ such that $A^{32} = [01 - 1 0]$ . Then a value of $α$ is

(A)

\frac{π}{16}

(B)

0

(C)

\frac{π}{32}

(D)

\frac{π}{64}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=​12a​212​2−2b​​ is a matrix satisfying the equation AAT=9I, where I is 3×3 identity matrix, then the ordered pair (a,b) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A is a 3×3 non-singular matrix such that AA′=A′A and B=A−1A′, then BB′ equals:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​123​012​001​​. If u1​ and u2​ are column matrices such that Au1​=​100​​ and Au2​=​010​​, then u1​+u2​ is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A and B are square matrices of size n×n such that A2−B2=(A−B)(A+B), then which of the following will be always true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=[13​24​] and B=[a0​0b​], a,b∈N. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[11​01​] and I=[10​01​], then which one of the following holds for all n≥1, by the principle of mathematical induction

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[ab​ba​] and A2=[αβ​βα​], then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=​1416​014​001​​ and I be the identity matrix of order 3. If Q=[qij​] is a matrix such that P50−Q=I, then q21​q31​+q32​​ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a complex cube root of unity with ω=1 and P=[pij​] be an n×n matrix with pij​=ωi+j. Then P2=0, when n=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let M and N be two 3×3 non-singular skew-symmetric matrices such that MN=NM. If PT denotes the transpose of P, then M2N2(MTN)−1(MN−1)T is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P=[23​​−21​​21​23​​​] and A=[10​11​] and Q=PAPT and x=PTQ2005P then x is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[α1​01​] and B=[15​01​], then value of α for which A2=B, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A and B are square matrices of equal degree, then which one is correct among the followings?

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a matrix such that A⋅[10​23​] is a scalar matrix and ∣3A∣=108. Then A2 equals :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose A is any 3×3 non-singular matrix and (A−3I)(A−5I)=0, where I=I3​ and O=O3​. If αA+βA−1=4I, then α+β is equal to :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​111​011​001​​ and B=A20. Then the sum of the elements of the first column of B is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[cosθsinθ​−sinθcosθ​], then the matrix A−50 when θ=12π​, is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=​139​013​001​​ and Q=[qij​] be two 3×3 matrices such that Q−P5=I3​. Then q32​q21​+q31​​ is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=[cosαsinα​−sinαcosα​],(α∈R) such that A32=[01​−10​]. Then a value of α is

If $A = 12 a 212 2 - 2 b$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to:

If $A$ is a $3 \times 3$ non-singular matrix such that $A A^{'} = A^{'} A$ and $B = A^{- 1} A^{'}$ , then $B B^{'}$ equals:

Let $A = 123012001$ . If $u_{1}$ and $u_{2}$ are column matrices such that $A u_{1} = 100$ and $A u_{2} = 010$ , then $u_{1} + u_{2}$ is equal to:

If $A$ and $B$ are square matrices of size $n \times n$ such that $A^{2} - B^{2} = (A - B) (A + B)$ , then which of the following will be always true?

Let $A = [1324]$ and $B = [a 0 0 b]$ , $a, b \in N$ . Then

If $A = [1101]$ and $I = [1001]$ , then which one of the following holds for all $n \geq 1$ , by the principle of mathematical induction

If $A = [a b b a]$ and $A^{2} = [α β β α]$ , then

Let $P = 1416014001$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$ , then $\frac{q _{31} + q _{32}}{q _{21}}$ equals

Let $ω$ be a complex cube root of unity with $ω \neq = 1$ and $P = [p_{ij}]$ be an $n \times n$ matrix with $p_{ij} = ω^{i + j}$ . Then $P^{2} \neq = 0$ , when $n =$

Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N = N M$ . If $P^{T}$ denotes the transpose of $P$ , then $M^{2} N^{2} (M^{T} N)^{- 1} (M N^{- 1})^{T}$ is equal to

If $P = [\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2}]$ and $A = [1011]$ and $Q = P A P^{T}$ and $x = P^{T} Q^{2005} P$ then $x$ is equal to

If $A = [α 1 01]$ and $B = [1501]$ , then value of $α$ for which $A^{2} = B$ , is

If $A$ and $B$ are square matrices of equal degree, then which one is correct among the followings?

Let A be a matrix such that $A \cdot [1023]$ is a scalar matrix and $∣3 A ∣ = 108$ . Then $A^{2}$ equals :

Suppose A is any $3 \times 3$ non-singular matrix and $(A - 3 I) (A - 5 I) = 0$ , where $I = I_{3}$ and $O = O_{3}$ . If $α A + β A^{- 1} = 4 I$ , then $α + β$ is equal to :-

Let $A = 111011001$ and $B = A^{20}$ . Then the sum of the elements of the first column of B is :

If $A = [cos θ sin θ - sin θ cos θ]$ , then the matrix $A^{- 50}$ when $θ = \frac{π}{12}$ , is equal to :

Let $P = 139013001$ and $Q = [q_{ij}]$ be two $3 \times 3$ matrices such that $Q - P^{5} = I_{3}$ . Then $\frac{q _{21} + q _{31}}{q _{32}}$ is equal to:

Let $A = [cos α sin α - sin α cos α], (α \in R)$ such that $A^{32} = [01 - 1 0]$ . Then a value of $α$ is