Let A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 whose entries are real numbers. Which of the following statements is ALWAYS TRUE?

If A is skew-symmetric, then a 12 + a 11 + a 21 = 0

If A is symmetric, then a 21 + a 21 = 0

If A is symmetric, then a i i = 0 for all possible values of i

If A is symmetric, then a i i is positive for all possible value of i

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

If the matrix A = 0 a - 3 2 0 - 1 b 1 0 is skew symmetric, find the values of a and b .

Given that, A = 0 a - 3 2 0 - 1 b 1 0 And the matrix A is given to be skew-symmetric matrix ∴ A T = - A ∴ 0 2 b a 0 1 - 3 - 1 0 = - 0 a - 3 2 0 - 1 b 1 0 ⇒ 0 2 b a 0 1 - 3 - 1 0 = 0 - a 3 - 2 0 1 - b - 1 0 Two matrices A and B are said to be equal if they are of same order and their corresponding elements are equal. Therefore, - a = 2 and b = 3 ⇒ a = - 2 and b = 3 . Hence, the required values are: a = - 2 and b = 3

Show that A + A ' is symmetric matrix, if A = 2 4 3 5

Given, A = 2 4 3 5 We need to find: A + A ' We know that A ' is the notation for Transpose of A . And, transpose of a matrix is obtained by interchanging the rows and columns of the original matrix. Thus, A ' = 2 3 4 5 Now, A + A ' = 2 4 3 5 + 2 3 4 5 = 2 + 2 4 + 3 3 + 4 5 + 5 = 4 7 7 10 . . . . i Now, let us find : A + A ' ' A + A ' ' = 4 7 7 10 . . . . . i i Since A + A ' ' = A + A ' Hence, A + A ' is a symmetric matrix.

If A = 5 a b 0 and A is symmetric matrix, show that a = b .

Given that A = 5 a b 0 and A is symmetric matrix. A square matrix that is equal to its transpose is called a symmetric matrix. For example, a square matrix A = a i j is symmetric if and only if a i j = a j i for all values of i and j Now, transpose of A is A ' = 5 b a 0 According to the definition of symmetric matrix, A = A ' ⇒ 5 a b 0 = 5 b a 0 On equating we have a = b Hence, proved.

MEDIUM

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Let $A$ and $B$ be $3 \times 3$ matrices. Consider the following statements:
(I) If $A$ and $B$ are diagonal, then so is $A B$
(II) If $A$ and $B$ are symmetric, then so is $A B$

50% studentsanswered this correctly

Important Questions on Matrices and Determinants

EASY

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

Let

A = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]

whose entries are real numbers. Which of the following statements is ALWAYS 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, B

be invertible

2 \times 2

matrices such that

A

is symmetric and

B

is skew-symmetric. Let

C = A^{- 1} B^{- 1} + B^{- 1} A^{- 1}

and

D = B^{- 1} A B .

Then

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 ?

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

Let

A

and

B

be any two

3 \times 3

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

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:

EASY

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

A square matrix

A

is said to be skew-symmetric, if

A^{'} =

_____

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

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

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

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}

EASY

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

If the matrix

A = [\begin{matrix} 0 & a & - 3 \\ 2 & 0 & - 1 \\ b & 1 & 0 \end{matrix}]

is skew symmetric, find the values of

a

and

b

MEDIUM

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

Show that

(A + A^{'})

is symmetric matrix, if

A = (\begin{array}{l} 2 & 4 \\ 3 & 5 \end{array})

EASY

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

A = (\begin{matrix} 5 & a \\ b & 0 \end{matrix})

and

A

is symmetric matrix, show that

a = b

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:

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

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

Let A and B be 3×3 matrices. Consider the following statements: (I) If A and B are diagonal, then so is AB (II) If A and B are symmetric, then so is AB

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

Let $A$ and $B$ be $3 \times 3$ matrices. Consider the following statements:
(I) If $A$ and $B$ are diagonal, then so is $A B$
(II) If $A$ and $B$ are symmetric, then so is $A B$