Let A , be a 3 × 3 matrix, such that A 2 - 5 A + 7 I = O . Statement - I : A - 1 = 1 7 5 I - A . Statement - II : The polynomial A 3 - 2 A 2 - 3 A + I ,can be reduced to 5 A - 4 I . Then :

Statement - I is true, but Statement - II is false

Statement - I is false, but Statement - II is true

Let x ∈ R and let P = 1 1 1 0 2 2 0 0 3 , Q = 2 x x 0 4 0 x x 6 and R = P Q P - 1 . Then which of the following options is/are correct?

For x = 0 , if R 1 a b = 6 1 a b , then a + b = 5

For x = 1 , there exists a unit vector α i ^ + β j ^ + γ k ^ for which R α β γ = 0 0 0

There exists a real number x such that P Q = Q P

Let M and N be two 3 × 3 matrices such that M N = N M . Further, if M ≠ N 2 and M 2 = N 4 , then

There is a 3 × 3 non - zero matrix U such that M 2 + M N 2 U is the zero matrix

Determinant of M 2 + M N 2 ≥ 1

For a 3 × 3 matrix U , if M 2 + M N 2 U equals the zero matrix then U is the zero matrix

Let P 1 = I = 1 0 0 0 1 0 0 0 1 , P 2 = 1 0 0 0 0 1 0 1 0 , P 3 = 0 1 0 1 0 0 0 0 1 , P 4 = 0 1 0 0 0 1 1 0 0 , P 5 = 0 0 1 1 0 0 0 1 0 , P 6 = 0 0 1 0 1 0 1 0 0 , and X = ∑ k = 1 6 P K 2 1 3 1 0 2 3 2 1 P K T where P K T denotes the transpose of the matrix P K . Then which of the following options is/are correct?

X - 30 I is an invertible matrix

If 0 2 β γ α β − γ α − β γ is orthogonal, then the values of α , β and γ will be

α = ± 1 2 , β = ± 1 6 , γ = ± 1 3

α = ± 1 5 , β = ± 1 3 , γ = ± 1 2

α = ± 1 6 , β = ± 1 7 , γ = ± 1 3

α = ± 1 3 , β = ± 1 3 , γ = ± 1 3

EASY

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If $A$ is an orthogonal matrix, then $A^{- 1}$ equals

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Important Questions on Matrices and Determinants

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If $B$ is a $3 \times 3$ matrix such that $B^{2} = 0$ , then $d e t . [{(I + B)}^{50} - 50 B]$ is equal to :

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

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

and

B = A^{20}

. Then the sum of the elements of the first column of

B

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

ω

be a complex cube root of unity with

ω \neq 1

and

P = [p_{i j}]

be a

n \times n

matrix with

p_{i j} = ω^{i + j}

. Then

P^{2} \neq 0

, when

n =

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

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

and

Q = P A P^{T},

then

P^{T} Q^{2015} P

is :

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

A = [\begin{array}{l} x & y & z \\ y & z & x \\ z & x & y \end{array}],

where

x, y

and

z

are real numbers such that

x + y + z > 0

and

x y z = 2

. If

A^{2} = I_{3}

, then the value of

x^{3} + y^{3} + z^{3}

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

A

, be a

3 \times 3

matrix, such that

A^{2} - 5 A + 7 I = O .

Statement - I :

A^{- 1} = \frac{1}{7} (5 I - A) .

Statement - II : The polynomial

A^{3} - 2 A^{2} - 3 A + I

,can be reduced to

5 (A - 4 I) .

Then :

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A

and

B

are square matrices of the same order such that

(A + B) (A - B) =

A^{2} - B^{2},

than

{(A B A^{- 1})}^{2} i s e q u a l t o

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

x \in R

and let

P = [\begin{matrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{matrix}], Q = [\begin{matrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 6 \end{matrix}]

and

R = P Q P^{- 1} .

Then which of the following options is/are correct?

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

P = [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}]

and

I

be the identity matrix of order

3

. If

Q = [q_{i j}]

is a matrix such that

P^{50} - Q = I,

then the value of

\frac{q_{31} + q_{32}}{q_{21}}

is equal to

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

P = [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{matrix}]

and

Q = [q_{i j}]

be two

3 \times 3

matrices such that

Q - P^{5} = I_{3}

. Then

\frac{q_{21} + q_{31}}{q_{32}}

is equal to :

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Which of the following is (are) NOT the square of a

3 \times 3

matrix with real entries?

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A

is an orthogonal matrix, then

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}] \dots . [\begin{matrix} 1 & n - 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 78 \\ 0 & 1 \end{matrix}],

then the inverse of

[\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}]

is:

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

M

and

N

be two

3 \times 3

matrices such that

M N = N M

. Further, if

M \neq N^{2}

and

M^{2} = N^{4}

, then

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

P_{1} = I = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], P_{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}], P_{3} = [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}],

P_{4} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}], P_{5} = [\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}], P_{6} = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}],

and

X = \sum_{k = 1}^{6} P_{K} [\begin{matrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{matrix}] P_{K}^{T}

where

P_{K}^{T}

denotes the transpose of the matrix

P_{K} .

Then which of the following options is/are correct?

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A

is a symmetric matrix and

B

is skew- symmetric matrix such that

A + B = [\begin{matrix} 2 & 3 \\ 5 & - 1 \end{matrix}]

, then

A B

is equal to:

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

A = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}], (a \in R)

such that

A^{32} = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] .

Then, a value of

α

is:

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

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

, then which one of the following statements is not correct?

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

[\begin{array}{l} 0 2 β γ \\ α β - γ \\ α - β γ \end{array}]

is orthogonal, then the values of

α, β

and

γ

will be

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A = [\begin{array}{l} p & q & r \\ r & p & q \\ q & r & p \end{array}]

and

A A^{T} = I

, then

p^{3} + q^{3} + r^{3} =

If A is an orthogonal matrix, then A-1 equals

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

If $A$ is an orthogonal matrix, then $A^{- 1}$ equals