Adjoint and Inverse of a Matrix

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JEE Main 2016

If $A = [5 a 3 - b 2]$ and $A adj A = A A^{T}$ , then $5 a + b$ is equal to:

4

(B)

13

(C)

- 1

(D)

5

JEE Main 2013

LEVELBoard

If $P = 112 α 34 334$ is the adjoint of a $3 \times 3$ matrix $A$ and $∣ A ∣ = 4$ , then $α$ is equal to:

(A)

4

(B)

11

(C)

5

(D)

0

JEE Main 2009

LEVELJEE Main

Let $A$ be a $2 \times 2$ matrix. \\ Statement-1: $a d j (a d j A) = A$ \\ Statement-2: $∣ a d j A ∣ = ∣ A ∣$

(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

LEVELBoard

Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true?

(A)

d e t A = \pm 1

, then

A^{- 1}

exists but all its entries are not necessarily integers

(B)

d e t A \neq = \pm 1

, then

A^{- 1}

exists and all its entries are non integers

(C)

d e t A = \pm 1

, then

A^{- 1}

exists but all its entries are integers

(D)

d e t A = \pm 1

, then

A^{- 1}

need not exist

JEE Main 2005

LEVELBoard

If $A^{2} - A + I = 0$ , then the inverse of $A$ is

(A)

A + I

(B)

A

(C)

A - I

(D)

I - A

JEE Main 2004

LEVELBoard

Let $A = 121 - 1 11 1 - 3 1$ and $10 B = 4 - 5 1 20 - 2 2 α 3$ . If $B$ is the inverse of matrix $A$ , then $α$ is

(A)

5

(B)

- 1

(C)

2

(D)

- 2

JEE Advanced 2016

LEVELJEE Main

Let $P = 323 - 1 0 - 5 - 2 α 0$ , where $α \in R$ . Suppose $Q = [q_{ij}]$ is a matrix such that $P Q = k I$ , where $k \in R, k \neq = 0$ and $I$ is the identity matrix of order $3$ . If $q_{23} = - k /8$ and $det (Q) = k^{2} /2$ , then

* Multiple Correct Options

(A)

α = 0, k = 8

(B)

4 α - k + 8 = 0

(C)

det (P adj (Q)) = 2^{9}

(D)

det (Q adj (P)) = 2^{13}

JEE Advanced 2014

LEVELJEE Main

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if

* Multiple Correct Options

(A)

The first column of

M

is the transpose of the second row of

M

(B)

The second row of

M

is the transpose of the first column of

M

(C)

M

is a diagonal matrix with non-zero entries in the main diagonal

(D)

The product of entries in the main diagonal of

M

is not the square of an integer

JEE Advanced 2013

LEVELJEE Main

For $3 \times 3$ matrices $M$ and $N$ , which of the following statement(s) is (are) NOT correct?

* Multiple Correct Options

(A)

N^{T} M N

is symmetric or skew symmetric, according as

M

is symmetric or skew symmetric

(B)

M N - N M

is skew symmetric for all symmetric matrices

M

and

N

(C)

M N

is symmetric for all symmetric matrices

M

and

N

(D)

(adj M) (adj N) = adj (M N)

for all invertible matrices

M

and

N

JEE Advanced 2012

LEVELBoard

If the adjoint of a $3 \times 3$ matrix $P$ is $121411473$ , then the possible value(s) of the determinant of $P$ is (are)

* Multiple Correct Options

(A)

- 2

(B)

- 1

(C)

1

(D)

2

JEE Advanced 2005

LEVELJEE Main

$A = 100 01 - 2 014$ and $I = 100010001$ and $A^{- 1} = [\frac{1}{6} (A^{2} + c A + d I)]$ , then the value of $c$ and $d$ are

(A)

(a)

(- 6, - 11)

(B)

(b)

(6, 11)

(C)

(c)

(- 6, 11)

(D)

(d)

(6, - 11)

JEE Advanced 2010

LEVELJEE Main

Let $k$ be a positive real number and let $A = 2 k - 1 2 k - 2 k 2 k 1 2 k 2 k - 2 k - 1$ and $B = 0 1 - 2 k - k 2 k - 1 0 - 2 k k 2 k 0$ . If $det (adj A) + det (adj B) = 1 0^{6}$ , then $[k]$ is equal to [Note : adj M denotes the adjoint of square matrix M and [k] denotes the largest integer less than or equal to k.]

JEE Advanced 2006

LEVELJEE Main

Comprehension Passage

Let

A = 123012001

, and

U_{1}, U_{2}

and

U_{3}

are columns of a

3 \times 3

matrix

U

. If column matrices

U_{1}, U_{2}

and

U_{3}

satisfying

A U_{1} = 100, A U_{2} = 230, A U_{3} = 231

evaluate as directed in the following questions.

Question 1:

[72 - 63 - 84 51]

(B)

[51846372]

(C)

[51638472]

(D)

[72 - 84 - 63 51]

JEE Main 2019 (9 January)

LEVELJEE Main

If $A = e^{t} e^{t} e^{t} e^{- t} cos t - e^{- t} cos t - e^{- t} sin t 2 e^{- t} sin t e^{- t} sin t - e^{- t} sin t + e^{- t} cos t - 2 e^{- t} cos t$ Then A is-

(A)

Invertible only if

t = π /2

(B)

not invertible for any

t \in R

(C)

invertible for all

t \in R

(D)

invertible only if

t = π

JEE Main 2019 (12 April Shift 1)

LEVELJEE Main

If $B = 50 α 223 11 - 1$ is the inverse of a $3 \times 3$ matrix A, then the sum of all values of $α$ for which $det (A) + 1 = 0$ , is :

(A)

0

(B)

2

(C)

1

(D)

- 1

JEE Main 2020 (8 January Shift 2)

LEVELBoard

If $A = [2924]$ and $I = [1001]$ , then $10 A^{- 1}$ is equal to:

(A)

6 I - A

(B)

A - 6 I

(C)

4 I - A

(D)

A - 4 I

JEE Main 2020 (9 January Shift 1)

LEVELJEE Main

If the matrices $A = 111 13 - 1 243$ , $B = adj A$ and $C = 3 A$ , then $\frac{∣ adj B ∣}{∣ C ∣}$ is equal to:

(A)

16

(B)

2

(C)

8

(D)

72

JEE Main 2020 - 8 Jan (Evening)

LEVELJEE Main

If $A = (2924)$ and $I = (1001)$ , then $10 A^{- 1}$ is equal to :

(A)

4 I - A

(B)

6 I - A

(C)

A - 4 I

(D)

A - 6 I

JEE Main 2020 - 9 Jan (Morning)

LEVELJEE Main

If $A = 111 13 - 1 243$ , $B = a d j (A)$ and $C = 3 A$ then $\frac{∣ a d j B ∣}{∣ C ∣}$ is

JEE Main 2020 - 3 Sep (Evening)

LEVELJEE Main

Let $A$ be a $3 \times 3$ matrix such that $a d j A = 2 - 1 1 - 1 0 - 2 12 - 1$ and $B = a d j (a d j A)$ If $∣ A ∣ = λ$ and $∣ (B^{- 1})^{T} ∣ = μ$ , then the ordered pair, $(∣ λ ∣, μ)$ is equal to

(A)

(3,81)

(B)

(9, \frac{1}{9})

(C)

(3, \frac{1}{81})

(D)

(9, \frac{1}{81})

JEE Main 2021 (01 Sep Shift 2)

LEVELJEE Main

Let $J_{n, m} = \int_{0}^{1/2} \frac{x ^{n}}{x ^{m} - 1} d x, \forall n > m$ and $n, m \in N$ . Consider a matrix $A = [a_{ij}]_{3 \times 3}$ where $a_{ij} = {J_{6 + i, 3} - J_{i + 3, 3}, 0, i \leq j i > j$ . Then $∣ adj A^{- 1} ∣$ is :

(A)

(15)^{2} \times 2^{42}

(B)

(15)^{2} \times 2^{34}

(C)

(105)^{2} \times 2^{38}

(D)

(105)^{2} \times 2^{36}

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Main

Let $A$ be a $3 \times 3$ real matrix. If $det (2 Adj (2 Adj (Adj (2 A)))) = 2^{41}$ , then the value of $det (A^{2})$ equal .

JEE Main 2021 (20 July Shift 2)

LEVELJEE Main

Let $A = {a_{ij}}$ be a $3 \times 3$ matrix, where $a_{ij} = ⎩ ⎨ ⎧ (- 1)^{j - i} 2 (- 1)^{i + j} if i < j if i = j if i > j$ then $det (3 Adj (2 A^{- 1}))$ is equal to

JEE Main 2021 (27 July Shift 1)

LEVELJEE Main

Let $A = [1 - 1 24]$ . If $A^{- 1} = α I + β A, α, β \in R, I$ is a $2 \times 2$ identity matrix, then $4 (α - β)$ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[5a3​−b2​] and A adj A=AAT, then 5a+b is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P=​112​α34​334​​ is the adjoint of a 3×3 matrix A and ∣A∣=4, then α 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 2×2 matrix. \\ Statement-1: adj(adjA)=A \\ Statement-2: ∣adjA∣=∣A∣

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A2−A+I=0, then the inverse of A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​121​−111​1−31​​ and 10B=​4−51​20−2​2α3​​. If B is the inverse of matrix A, then α is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let M be a 2×2 symmetric matrix with integer entries. Then M is invertible if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For 3×3 matrices M and N, which of the following statement(s) is (are) NOT correct?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the adjoint of a 3×3 matrix P is ​121​411​473​​, then the possible value(s) of the determinant of P is (are)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A=​100​01−2​014​​ and I=​100​010​001​​ and A−1=[61​(A2+cA+dI)], then the value of c and d are

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value ∣U∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the elements of the matrix U−1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of [3​2​0​]U​320​​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[2−4​−31​], then adj(3A2+12A) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=​etetet​e−tcost−e−tcost−e−tsint2e−tsint​e−tsint−e−tsint+e−tcost−2e−tcost​​ Then A is-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If B=​50α​223​11−1​​ is the inverse of a 3×3 matrix A, then the sum of all values of α for which det(A)+1=0, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[29​24​] and I=[10​01​], then 10A−1 is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the matrices A=​111​13−1​243​​, B=adj A and C=3A, then ∣C∣∣adj B∣​ is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=(29​24​) and I=(10​01​), then 10A−1 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=​111​13−1​243​​, B=adj(A) and C=3A then ∣C∣∣adjB∣​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Jn,m​=∫01/2​xm−1xn​dx,∀n>m and n,m∈N. Consider a matrix A=[aij​]3×3​ where aij​={J6+i,3​−Ji+3,3​,0,​i≤ji>j​. Then ∣adj A−1∣ is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a 3×3 real matrix. If det(2 Adj(2 Adj(Adj(2A))))=241, then the value of det(A2) equal .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={aij​} be a 3×3 matrix, where aij​=⎩⎨⎧​(−1)j−i2(−1)i+j​if i<jif i=jif i>j​ then det(3Adj(2A−1)) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=[1−1​24​]. If A−1=αI+βA,α,β∈R,I is a 2×2 identity matrix, then 4(α−β) is equal to :

If $A = [5 a 3 - b 2]$ and $A adj A = A A^{T}$ , then $5 a + b$ is equal to:

If $P = 112 α 34 334$ is the adjoint of a $3 \times 3$ matrix $A$ and $∣ A ∣ = 4$ , then $α$ is equal to:

Let $A$ be a $2 \times 2$ matrix. \\ Statement-1: $a d j (a d j A) = A$ \\ Statement-2: $∣ a d j A ∣ = ∣ A ∣$

Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true?

If $A^{2} - A + I = 0$ , then the inverse of $A$ is

Let $A = 121 - 1 11 1 - 3 1$ and $10 B = 4 - 5 1 20 - 2 2 α 3$ . If $B$ is the inverse of matrix $A$ , then $α$ is

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if

For $3 \times 3$ matrices $M$ and $N$ , which of the following statement(s) is (are) NOT correct?

If the adjoint of a $3 \times 3$ matrix $P$ is $121411473$ , then the possible value(s) of the determinant of $P$ is (are)

$A = 100 01 - 2 014$ and $I = 100010001$ and $A^{- 1} = [\frac{1}{6} (A^{2} + c A + d I)]$ , then the value of $c$ and $d$ are

The value $∣ U ∣$ is

The sum of the elements of the matrix $U^{- 1}$ is

The value of $[320] U 320$ is

If $A = [2 - 4 - 3 1]$ , then $adj (3 A^{2} + 12 A)$ is equal to:

If $A = e^{t} e^{t} e^{t} e^{- t} cos t - e^{- t} cos t - e^{- t} sin t 2 e^{- t} sin t e^{- t} sin t - e^{- t} sin t + e^{- t} cos t - 2 e^{- t} cos t$ Then A is-

If $B = 50 α 223 11 - 1$ is the inverse of a $3 \times 3$ matrix A, then the sum of all values of $α$ for which $det (A) + 1 = 0$ , is :

If $A = [2924]$ and $I = [1001]$ , then $10 A^{- 1}$ is equal to:

If the matrices $A = 111 13 - 1 243$ , $B = adj A$ and $C = 3 A$ , then $\frac{∣ adj B ∣}{∣ C ∣}$ is equal to:

If $A = (2924)$ and $I = (1001)$ , then $10 A^{- 1}$ is equal to :

If $A = 111 13 - 1 243$ , $B = a d j (A)$ and $C = 3 A$ then $\frac{∣ a d j B ∣}{∣ C ∣}$ is

Let $J_{n, m} = \int_{0}^{1/2} \frac{x ^{n}}{x ^{m} - 1} d x, \forall n > m$ and $n, m \in N$ . Consider a matrix $A = [a_{ij}]_{3 \times 3}$ where $a_{ij} = {J_{6 + i, 3} - J_{i + 3, 3}, 0, i \leq j i > j$ . Then $∣ adj A^{- 1} ∣$ is :

Let $A$ be a $3 \times 3$ real matrix. If $det (2 Adj (2 Adj (Adj (2 A)))) = 2^{41}$ , then the value of $det (A^{2})$ equal .

Let $A = {a_{ij}}$ be a $3 \times 3$ matrix, where $a_{ij} = ⎩ ⎨ ⎧ (- 1)^{j - i} 2 (- 1)^{i + j} if i < j if i = j if i > j$ then $det (3 Adj (2 A^{- 1}))$ is equal to

Let $A = [1 - 1 24]$ . If $A^{- 1} = α I + β A, α, β \in R, I$ is a $2 \times 2$ identity matrix, then $4 (α - β)$ is equal to :