Let M = 0 1 a 1 2 3 3 b 1 and a d j M = - 1 1 - 1 8 - 6 2 - 5 3 - 1 where a and b are real numbers. Which of the following options is/are correct?

If M = α β γ = 1 2 3 , then α - β + γ = 3

MEDIUM

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Let $B = [\begin{array}{l} 1 & 3 & α \\ 1 & 2 & 3 \\ α & α & 4 \end{array}], α > 2$ be the adjoint of a matrix $A$ and $|A| = 2 .$ Then $[\begin{matrix} α & - 2 α & α \end{matrix}] B [\begin{matrix} α \\ - 2 α \\ α \end{matrix}]$ is equal to

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Important Questions on Matrices and Determinants

HARD

Mathematics>Algebra>Matrices and Determinants>Adjoint 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>Matrices and Determinants>Adjoint 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>Matrices and Determinants>Adjoint 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)

= ….

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Adjoint 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>Matrices and Determinants>Adjoint 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

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Adjoint 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

MEDIUM

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

Let

M = [\begin{matrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{matrix}]

and

a d j M = [\begin{matrix} - 1 & 1 & - 1 \\ 8 & - 6 & 2 \\ - 5 & 3 & - 1 \end{matrix}]

where

a

and

b

are real numbers. Which of the following options is/are correct?

HARD

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

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

, then

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

is equal to:

MEDIUM

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

For an invertible matrix

A

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

then

| A | =

EASY

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

A

is a square matrix of order

3

and

| A | = 5

, then

|A \cdot a d j A|

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Adjoint 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>Matrices and Determinants>Adjoint 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

EASY

Mathematics>Algebra>Matrices and Determinants>Adjoint 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>Matrices and Determinants>Adjoint 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>Matrices and Determinants>Adjoint 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>Matrices and Determinants>Adjoint 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

MEDIUM

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

For

3 \times 3

matrices

M

and

N

, which of the following statement(s) is (are) NOT correct?

HARD

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

A

is a

3 \times 3

matrix such that

|5 a d j A| = 5

, then

|A|

is equal to

HARD

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

Let

P = [\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}]

, where

α \in R,

Suppose

Q = [q_{i j}]

is a matrix such that

P Q = k I

, where

k \in R, k \neq0

and

I

is the identity matrix of order

3

. If

q_{23} = - \frac{k}{8}

and

\det (Q) = \frac{k^{2}}{2}

then

Let B=13α123αα4,α>2 be the adjoint of a matrix A and A=2. Then α-2ααBα-2αα is equal to

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

Let $B = [\begin{array}{l} 1 & 3 & α \\ 1 & 2 & 3 \\ α & α & 4 \end{array}], α > 2$ be the adjoint of a matrix $A$ and $|A| = 2 .$ Then $[\begin{matrix} α & - 2 α & α \end{matrix}] B [\begin{matrix} α \\ - 2 α \\ α \end{matrix}]$ is equal to