Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

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If $A B = I$ and $B = A^{'}$ then :

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Important Questions on Matrices and Determinants

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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

If

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

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Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Inverse of a diagonal non-singular matrix 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}

is

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If

A = [\begin{matrix} e^{t} & e^{- t} c o s t & e^{- t} \sin t \\ e^{t} & - e^{- t} \cos t - e^{- t} \sin t & - e^{- t} \sin t + e^{- t} \cos t \\ e^{t} & {2 e}^{- t} \sin t & - 2 e^{- t} \cos t \end{matrix}],

then

A

is:

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Inverse of

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

is

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Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If

A = [\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix}]

and

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

then

10 A^{- 1}

, is equal to.

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If

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

If

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

HARD

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

The inverse of the matrix

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

is

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Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If

A

is an orthogonal matrix, then

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 :

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 :

EASY

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

If

A

is a non-singular matrix and

(A + I) (A - I) = 0

then

A + A^{- 1} = \dots \dots

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Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

The inverse of matrix is

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

is

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Algebra of Matrices

Let

M = [\begin{matrix} \sin^{4} θ & - 1 - \sin^{2} θ \\ 1 + \cos^{2} θ & \cos^{4} θ \end{matrix}] = α I + β M^{- 1},

where

α = α (θ)

and

β = β (θ)

are real number, and

I

is the

2 \times 2

identity matrix. If

α^{*}

is the minimum of the set {

α (θ) : θ \in [0, 2 π)}

and

β^{*}

is the minimum of the set

\{β (θ) : θ \in [0, 2 π)\},

then the value of

α^{*} + β^{*}

is

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

If

[\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

If

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} =