Let A = 0 - 2 2 0 . If M and N are two matrices given by M = ∑ k = 1 10 A 2 k and N = ∑ k = 1 10 A 2 k - 1 then M N 2 is

a non-identity symmetric matrix

neither symmetric nor skew-symmetric matrix

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Which of the following is not true?

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Important Questions on Matrices and Determinants

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Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

A

and

B

be any two

3 \times 3

symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

A

and

B

are symmetric matrices of same order and

X = A B + B A

and

Y = A B - B A

, then

{(X Y)}^{T}

is equal to

HARD

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

A = [\begin{array}{r} 0 & - 2 \\ 2 & 0 \end{array}]

. If

M

and

N

are two matrices given by

M = \sum_{k = 1}^{10} A^{2 k}

and

N = \sum_{k = 1}^{10} A^{2 k - 1}

then

M N^{2}

HARD

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

X a n d Y

be two arbitrary,

3 \times 3

, non - zero, skew - symmetric matrices and Z be an arbitrary

3 \times 3

, non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

P

and

Q

are symmetric matrices of the same order, then

P Q - Q P

HARD

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

A

be a symmetric matrix such that

|A| = 2

and

[\begin{array}{l} 2 & 1 \\ 3 & \frac{3}{2} \end{array}] A = [\begin{array}{l} 1 & 2 \\ α & β \end{array}]

If the sum of the diagonal elements of

A

s

, then

\frac{β s}{α^{2}}

is equal to _________.

HARD

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

A = [\begin{array}{l} 2 & 3 \\ a & 0 \end{array}], a \in R

be written as

P + Q

where

P

is a symmetric matrix and

Q

is skew symmetric matrix. If

det (Q) = 9

, then the modulus of the sum of all possible values of determinant of

P

is equal to:

HARD

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

|M|

denote the determinant of a square matrix

M

. Let

g : [0, \frac{π}{2}] \to ℝ

be the function defined by

g (θ) = \sqrt{f (θ) - 1} + \sqrt{f (\frac{π}{2} - θ) - 1}

where

f (θ) = \frac{1}{2} |\begin{matrix} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{matrix}| + |\begin{matrix} \sin π & \cos (θ + \frac{π}{4}) & \tan (θ - \frac{π}{4}) \\ \sin (θ - \frac{π}{4}) & - \cos \frac{π}{2} & \log_{e} (\frac{4}{π}) \\ \cot (θ + \frac{π}{4}) & \log_{e} (\frac{π}{4}) & \tan π \end{matrix}|

Let

p (x)

be a quadratic polynomial whose roots are the maximum and minimum values of the function

g (θ)

, and

p (2) = 2 - \sqrt{2}

. Then, which of the following is/are TRUE ?

EASY

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

A

and

B

are square matrices of same order and

B

is a skew symmetric matrix, then

A^{'} B A

EASY

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

f (x) = |\begin{matrix} x^{3} - x & a + x & b + x \\ x - a & x^{2} - x & c + x \\ x - b & x - c & 0 \end{matrix}|

then

HARD

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

M

be a

2 \times 2

symmetric matrix with integer entries. Then

M

is invertible if

EASY

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric 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>Symmetric and Skew-Symmetric Matrices

In a third order matrix $A, a_{i j}$ denotes the element in the $i^{th}$ row and $j^{th}$ column.

If $a_{i j} = 0$ for $i = j$
$= 1$ for $i > j$
$= - 1$ for $i < j$

Then the matrix is

EASY

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let $A, B, C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric.

Consider the statements

$(S 1) A^{13} B^{26} - B^{26} A^{13}$ is symmetric

$(S 2) A^{26} C^{13} - C^{13} A^{26}$ is symmetric

Then,

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

A

be a symmetric matrix of order

2

with integer entries. If the sum of the diagonal elements of

A^{2}

1,

then the possible number of such matrices is:

EASY

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Let

A

and

B

be any two

3 \times 3

matrices. If

A

is symmetric and

B

is skew symmetric, then the matrix

A B - B A

EASY

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

A, B

are symmetric matrices of the same order, then

A B - B A

is a

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

A

is a square matrix of order

3

with

|A| = 2,

then the value of

|{(A - A^{T})}^{5}| + |{(A^{T} - A)}^{3}|

HARD

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix

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

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Symmetric and Skew-Symmetric Matrices

A, B

are symmetric matrices of the same order then

(A B - B A)

is -

Which of the following is not true?

Important Questions on Matrices and Determinants

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