Find the adjoint of the matrix A = 1 2 3 - 5 and verify the result A ( adj A ) = ( adj A ) A = | A | · I .

Given that, the matrix is A = 1 2 3 - 5 Adjoint of a 2 × 2 matrix is obtained by interchanging diagonal elements and adding negative sign in front of other elements. ∴ Adj A = - 5 - 2 - 3 1 Now, A ( adj A ) = 1 2 3 - 5 - 5 - 2 - 3 1 = - 5 - 6 - 2 + 2 - 15 + 15 - 6 - 5 = - 11 0 0 - 11 Similarly, ( adj A ) A = - 5 - 2 - 3 1 1 2 3 - 5 = - 5 - 6 - 10 + 10 - 3 + 3 - 6 - 5 = - 11 0 0 - 11 Also, A · I = - 11 1 0 0 1 = - 11 0 0 - 11 Hence, A ( adj A ) = ( adj A ) A = | A | · I

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 $A = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$ , then which of the following statement is not correct?

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Important Questions on Matrices and Determinants

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Find the adjoint of the matrix $A = [\begin{array}{r} 1 & 2 \\ 3 & - 5 \end{array}]$ and verify the result $A (adj A) = (adj A) A = | A | \cdot I$ .

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

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

is the inverse of a

3 \times 3

matix

A,

then the sum of all values of

α

for which

d e t (A) + 1 = 0,

is:

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If $A$ is a scalar matrix with scalar $k \neq 0$ , of order $3$ then $A^{- 1}$ is

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 = [\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}

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

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Inverse of

[\begin{array}{l} 3 & 1 \\ 5 & 2 \end{array}]

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 ___________.

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A

is a

3 \times 3

matrix such that

|5 a d j A| = 5

, then

|A|

is equal to

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

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

and

B = [\begin{matrix} 3 \\ 1 \end{matrix}], A X = B

, then

X

is equal to

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

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

A

is an orthogonal matrix, then

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

A

is a scalar matrix with scalar

k \neq 0

, of order

3

, then

A^{- 1}

is :

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

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_{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?

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

A

be any

3 \times 3

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

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=cosθ-sinθsinθcosθ, then which of the following statement is not correct?

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

If $A = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$ , then which of the following statement is not correct?