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 A = cos α sin α - sin α cos α , then verify that A ' A = I

Given: A = cos α sin α - sin α cos α To get transpose interchange rows and columns. ⇒ A ' = cos α - sin α sin α cos α Therefore A ' A = cos α - sin α sin α cos α cos α sin α - sin α cos α = cos 2 α + sin 2 α cos α sin α - cos α sin α cos α sin α - cos α sin α cos 2 α + sin 2 α = 1 0 0 1 = I ∴ A ' A = I Hence, proved

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

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Find number of possible triplets $(α, β, γ)$ , if $A = |\begin{matrix} 0 & α & α \\ 2 β & β & - β \\ γ & - γ & γ \end{matrix}|$ is an orthogonal matrix.

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Important Questions on Matrices and Determinants

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?

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

For two

3 \times 3

matrices

A

and

B

, let

A + B = 2 B^{'}

and

3 A + 2 B = I_{3},

where

B^{'}

is the transpose of

B

and

I_{3}

3 \times 3

identity matrix. Then :

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

The number of all

3 \times 3

matrices

A,

with entries from the set

\{- 1, 0, 1\}

such that the sum of the diagonal elements of

A A^{T}

3,

is ___________.

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

P = [\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}]

a b c = 1

P^{T} P = I

then the value of

a^{3} + b^{3} + c^{3}

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}

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

3 A + 4 B^{'} = [\begin{matrix} 7 & - 10 & 17 \\ 0 & 6 & 31 \end{matrix}]

and

2 B - 3 A^{'} = [\begin{matrix} - 1 & 18 \\ 4 & 0 \\ - 5 & - 7 \end{matrix}]

then

B =

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A = [\begin{matrix} 5 a & - b \\ 3 & 2 \end{matrix}]

and

A . a d j A = A A^{T}

, then

5 a + b

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 :

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If $A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{matrix}]$ 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

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A

is a

3 \times 3

non-singular matrix such that

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

and

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

then

B B^{'}

equals, where

X^{'}

denotes the transpose of the matrix

X

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

A, B, C, D

be square real matrices such that

C^{T} = D A B, D^{T} = A B C, S = A B C D

, then

S^{2}

is equal to

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A

is an orthogonal matrix, then

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A = [\begin{array}{l} \cos α & \sin α \\ - \sin α & \cos α \end{array}]

, then verify that

A^{'} A = I

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

The total number of matrices

A = [\begin{matrix} 0 & 2 y & 1 \\ 2 x & y & - 1 \\ 2 x & - y & 1 \end{matrix}], (x, y \in R, x \neq y)

for which

A^{T} A = 3 I_{3}

is:

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

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

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?

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If the matrix

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

where

A

is symmetric and

B

is skew-symmetric then

B

is equal to

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} =

Find number of possible triplets α, β, γ, if A=0αα2ββ-βγ-γγ is an orthogonal matrix.

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

Find number of possible triplets $(α, β, γ)$ , if $A = |\begin{matrix} 0 & α & α \\ 2 β & β & - β \\ γ & - γ & γ \end{matrix}|$ is an orthogonal matrix.