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 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 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

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Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in ℕ$ , if $P^{α} + P^{β} = γ l - 29 P$ and $P^{α} - P^{β} = δ l - 13 P$ , then $α + β + γ - δ$ is equal to

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Important Questions on Matrices

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 :

HARD

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

MEDIUM

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

HARD

Mathematics>Algebra>Matrices>Operation on Matrices

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

3 \times 3

matrix with real entries?

MEDIUM

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 :

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

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

MEDIUM

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

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?

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 :

MEDIUM

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:

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

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:

EASY

Mathematics>Algebra>Matrices>Operation on Matrices

[\begin{matrix} - 1 & 3 \\ 4 & - 5 \\ 0 & 2 \end{matrix}] [\begin{matrix} 1 & 2 \\ 0 & 7 \end{matrix}] = [\begin{matrix} - 1 & 19 \\ α & - 27 \\ 0 & 14 \end{matrix}]

, then the value of

α

EASY

Mathematics>Algebra>Matrices>Operation on Matrices

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

and

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

are matrices, then

A B + B A

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

MEDIUM

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

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

Let P be a square matrix such that P2=I-P. For α, β, γ, δ∈ℕ, if Pα+Pβ=γl-29P and Pα-Pβ=δl-13P, then α+β+γ-δ is equal to

Important Questions on Matrices

Learn and Prepare for any exam you want !

Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in ℕ$ , if $P^{α} + P^{β} = γ l - 29 P$ and $P^{α} - P^{β} = δ l - 13 P$ , then $α + β + γ - δ$ is equal to