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Let $A = {[a_{i j}]}_{5 \times 5}$ is a matrix such that $a_{i j} = \{\begin{matrix} 3 & \forall i = j \\ 0 & \forall i \neq j \end{matrix}$ . If $|\frac{a d j (a d j A)}{3}|$ $= {(\sqrt{3})}^{λ}$ , then $λ$ is equal to (where, $a d j (M)$ represents the adjoint matrix of matrix $M$ )

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Important Questions on Determinants

HARD

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is a

3 \times 3

matrix such that

|5 a d j A| = 5

, then

|A|

is equal to

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

If the value of a third order determinant is

16

, then the value of the determinant formed by replacing each of its elements by its cofactor is

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is an

3 \times 3

non- singular matrix, such that

A A^{'} = A^{'} A

and

B = A^{- 1} A^{'}

, then

B B^{'}

equals to {where

A^{'}

transpose of matrix

A

}

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{array}{l} 1 & 2 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \end{array}],

then

{[adj (adj A)]}^{- 1} =

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

P = [\begin{matrix} 1 & α & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{matrix}]

is the adjoint of a

3 \times 3

matrix

A

and

|A| = 4

, then

α

is equal to

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is a

3 \times 3

matrix and the matrix obtained by replacing the elements of

A

with their corresponding cofactors is

[\begin{matrix} 1 & - 2 & 1 \\ 4 & - 5 & - 2 \\ - 2 & 4 & 1 \end{matrix}]

, then a possible value of the determinant of

A

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

Let

A

be a

3 \times 3

matrix such that

adj A = [\begin{matrix} 2 & - 1 & 1 \\ - 1 & 0 & 2 \\ 1 & - 2 & - 1 \end{matrix}]

and

B = adj (a d j A)

. If

| A | = λ

and

|{(B^{- 1})}^{T}| = μ

, then the ordered pair

(|λ|, μ)

is equal to

HARD

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} 2 & 3 \\ 0 & - 1 \end{matrix}],

then the value of

det (A^{4}) + det (A^{10} - (Adj (2 A))^{10})

is equal to ________.

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

Let

A

be any

3 \times 3

invertible matrix. Then which one of the following is not always true?

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} 1 + 2 i & i \\ - i & 1 - 2 i \end{matrix}]

, where

i = \sqrt{- 1}

, then

A (a d j A)

= ….

HARD

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

Match the items of List-

I

with the items of List-

II

and choose the correct option

List- $I$		List- $II$
$(A)$	If $A$ is a non-singular matrix of order $3$ and $\| A \| = a,$ then $\|{(adj A^{- 1})}^{- 1}\| =$	$(I)$	null matrix
$(B)$	A is a non-singular matrix of order $3$ and $B$ is any matrix of order $3$ such that $A B = O$ , then $B$ is	$(II)$	$a^{2}$
$(C)$	$\|\begin{matrix} 1 & x & x^{2} \\ \cos (a - b) y & \cos a y & \cos (a + b) y \\ \sin (a - b) y & \sin a y & \sin (a + b) y \end{matrix}\|$ does not depend on	$(III)$	$b$
$(D)$	$A$ is a square matrix of order $3$ and $B = A - A^{T}$ , then $\|B\|$ is	$(IV)$	$a$
		$(V)$	$0$

The correct answer is

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} 2 & - 3 \\ - 4 & 1 \end{matrix}]

, then

Adj (3 A^{2} + 12 A)

is equal to:

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

M

is any square matrix of order

3

over

ℝ

and if

M^{'}

be the transpose of

M,

then

adj (M^{'}) - (adj M)^{'}

is equal to

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is a square matrix of order

3

and

| A | = 5

, then

|A \cdot a d j A|

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{array}{l} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{array}]

, then

det (Adj A)

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

be a square matrix of order

n

and if

|A| = D

and

|a d j A| = D'

, then

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is a

3 \times 3

non-singular matrix such that

A A^{T} = A^{T} A

and

B = A^{- 1} A^{T}

, then

B B^{T}

is equal to:

HARD

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & - 1 & 3 \end{matrix}], B = a d j A

and

C = 3 A,

then

\frac{|a d j B|}{|C|}

is equal to

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

For an invertible matrix

A

A (a d j A) = [\begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix}]

then

| A | =

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

The characteristic equation of a matrix

A

λ^{3} - 5 λ^{2} - 3 λ + 2 I = 0

then

|a d j A|

is equal to

Let A=aij5×5 is a matrix such that aij=3∀i=j0∀i≠j. If adj(adjA)3=3λ, then λ is equal to (where, adj(M) represents the adjoint matrix of matrix M)

Important Questions on Determinants

Learn and Prepare for any exam you want !

Let $A = {[a_{i j}]}_{5 \times 5}$ is a matrix such that $a_{i j} = \{\begin{matrix} 3 & \forall i = j \\ 0 & \forall i \neq j \end{matrix}$ . If $|\frac{a d j (a d j A)}{3}|$ $= {(\sqrt{3})}^{λ}$ , then $λ$ is equal to (where, $a d j (M)$ represents the adjoint matrix of matrix $M$ )