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

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Let $A$ be $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr (A),$ the sum of diagonal entries of $A .$ Assume that $A^{2} = I .$

Statement-I: If $A \neq I$ and $A \neq - I,$ then $det A = - 1 .$

Statement-II : If $A \neq I$ and $A \neq - I,$ then $t r (A) \neq 0 .$

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Important Questions on Matrices

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Mathematics>Algebra>Matrices>Operation on 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 :

HARD

Mathematics>Algebra>Matrices>Operation on 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

HARD

Mathematics>Algebra>Matrices>Operation on 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 :

HARD

Mathematics>Algebra>Matrices>Operation on 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 :

EASY

Mathematics>Algebra>Matrices>Operation on 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>Operation on 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>Operation on Matrices

[\begin{matrix} 1 & x & 1 \end{matrix}] [\begin{matrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ x \end{matrix}] = 0

,

then

x

can be

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Mathematics>Algebra>Matrices>Operation on 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:

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Mathematics>Algebra>Matrices>Operation on Matrices

A

3 \times 4

matrix and

B

is a matrix such that

A^{'} B

and

B^{'} A

are both defined, then the order of

B

can be

MEDIUM

Mathematics>Algebra>Matrices>Operation on 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

EASY

Mathematics>Algebra>Matrices>Operation on 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 ___________.

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Mathematics>Algebra>Matrices>Operation on Matrices

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

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

HARD

Mathematics>Algebra>Matrices>Operation on 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:

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Mathematics>Algebra>Matrices>Operation on 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>Operation on 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:

EASY

Mathematics>Algebra>Matrices>Operation on Matrices

Let

α

be a root of the equation

x^{2} + x + 1 = 0

and the matrix

A = \frac{1}{\sqrt{3}} [\begin{matrix} 1 & 1 & 1 \\ 1 & α & α^{2} \\ 1 & α^{2} & α^{4} \end{matrix}],

then the matrix

A^{31}

is equal to

EASY

Mathematics>Algebra>Matrices>Operation on 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:

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Mathematics>Algebra>Matrices>Operation on 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>Operation on 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

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Mathematics>Algebra>Matrices>Operation on 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

Let A be 2×2 matrix with real entries. Let I be the 2×2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A2=I. Statement-I: If A≠I and A≠-I, then det A=-1. Statement-II : If A≠I and A≠-I, then tr(A)≠0.

Important Questions on Matrices

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