Symmetric and Non-symmetric Matrices MCQ Quiz - Objective Question with Answer for Symmetric and Non-symmetric Matrices - Download Free PDF

Calculation: 

Given, A^T = A, B^T = –B, C^T = –C

Let M = A¹³B²⁶ – B²⁶A¹³

⇒ Then, M^T = (A¹³B²⁶ – B²⁶A¹³)^T

= (A¹³B²⁶)T – (B²⁶A¹³)^T

= (B^T)²⁶(A^T)¹³ – (A^T)¹³(B^T)²⁶

= B²⁶A¹³ – A¹³ B²⁶ = –M

Hence, M is skew symmetric

Let, N = A²⁶C¹³ – C¹³A²⁶

⇒ then, N^T = (A²⁶C¹³)^T – (C¹³A²⁶)^T

= –(C)¹³(A)²⁶+ A²⁶C¹³ = N

Hence, N is symmetric.

∴ Only S2 is true.

Hence, the correct answer is Option 1.

Concept

(A + B)T = A + B (symmetric)

(A - B)T = - (A - B) (skew-symmetric)

Calculation

Given:

A + B is symmetric, A - B is skew-symmetric, D = CT

A = , C = 

⇒ D = C^T =  

A^T + B^T = A + B ...(1)

A^T - B^T = -A + B ...(2)

⇒ (1) + (2): 2A^T = 2B

⇒ B = A^T

⇒ B =

B + D = +

⇒ B + D =

⇒ B + D =

∴ B + D =

Hence option 2 is correct

Concept Used:

If A is a symmetric matrix, then A^T = A.

If B is a skew-symmetric matrix, then B^T = -B.

If C = A + B, then C^T = A^T + B^T.

Calculation:

Given:

A is a symmetric matrix.

B is a skew-symmetric matrix.

 A + B = ...(1)

⇒ (A + B)^T =

⇒ A^T + B^T =

⇒ A - B = ...(2)

Adding (1) and (2):

⇒ (A + B) + (A - B) = +

⇒ 2A =

⇒ A =

Subtracting (2) from (1):

⇒ (A + B) - (A - B) = -

⇒ 2B =

⇒ B =

A - B =

∴ A - B =

Hence option 4 is correct

Concept

Square matrix A is said to be symmetric if aij = aij for all i and j.

Square matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A ⇔ AT = A

Calculation

For a symmetric matrix, AT = A

  is symmetric as all aij = aij

∴ Option 2 is correct

Concept:

A square matrix A = [a_ij]_{n × n} is said to be symmetric if A^T = A

A^T (Transpose) is obtained by changing rows to columns and columns to rows

Calculation:

Let A = 

 A^T =  = A

 A is  a symmetric matrix

CONCEPT:

Symmetric Matrix:

Any real square matrix A = (a_ij) is said to be symmetric matrix if and only if a_ij = a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = A^t then A is said to be a symmetric matrix.

Skew-symmetric Matrix:

Any real square matrix A = (a_ij) is said to be skew-symmetric matrix if and only if a_ij = - a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = - A^t then A is said to be a skew-symmetric matrix.

Properties of Transpose of a Matrix:

• If A is a matrix of order m × n, then (A^t)^t = A
• If k ∈ R is a scalar and A is a matrix of order m × n, then (k × A)^t = k × A^t
• If A and B are matrices of same order m × n, then (A ± B)^t = A^t ± B^t.

CALCULATION:

Given: A is skew symmetric matrix

As we know that, if A is a skew symmetric matrix i.e A = - At 

⇒ (A²)^t = (A^t)²

∵ A is skew symmetric matrix

⇒ (A2)t = (- A)² = A²

So, A² is a symmetric matrix.

Hence, option D is the correct answer.

Concept:

Consider a matrix A is skew-symmetric, then A^T = −A
and A is symmetric, then A^T = A

Calculation:

Since, A is skew-symmetric.
A^T = −A
Since, A is symmetric.
A^T = A
⇒ −A = A
⇒2A = O
⇒A = O
Hence, A is a null matrix.

Hence, option (4) is correct.

Concept: 

A matrix X can be written as a sum of the symmetric and skew-symmetric matrix which are 

P(symmetric) = (X + XT)

Q(skew-symmetric) = (X - XT)

Where XT is transpose of matrix X

Given matrix X = 

Transpose matrix X^T = 

Symmetric matrix (P) =  (X + XT)

⇒ P =  

⇒ P =  

⇒ P = 

Skew-symmetric matrix (Q) = (X - XT)

⇒ Q =  

⇒ Q =  

⇒ Q = 

CONCEPT:

Symmetric Matrix:

Any real square matrix A = (aij) is said to be symmetric matrix if and only if aij = aji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = At then A is said to be a symmetric matrix.

Calculation:

A = At

On comparing 

3 + x = 1 - x

⇒ x = - 1

And, y + 1 = 5 - y

⇒ y = 2

3x + y = 3(-1) + 2

∴ 3x + y = -1 

Explanation:

Given, the matrix A =  

A = B + C

Here matrix A is expressed as the sum of symmetric and skew-symmetric matrices.

Then, 

Where B is symmetric and C is a skew-symmetric matrix.

To Find: The matrix B

A =  

Concept:

A matrix X can be written as a sum of symmetric and skew-symmetric matrix which are 

P(symmetric) = (X + X^T)

Q(skew-symmetric) = (X - XT)

Calculation:

A = 

A^T = 

P(symmetric matrix) = (A + AT)

⇒ P = 

⇒ P = 

Q(skew-symmetric) = (A - AT)

Concept:

• Symmetric Matrix: Any real square matrix A = (a_ij) is said to be symmetric matrix if and only if a_ij = a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = A’ then A is said to be a symmetric matrix.
• (A ± B)' = A' ± B'
• (A ⋅ B)' = B' ⋅ A'

Calculation:

Given: A and B are two symmetric matrices of order n

Statement 1: A + B is also a symmetric matrix.

Let's find out transpose of A + B

⇒ (A + B)' = A' + B'

∵ A and B are two symmetric matrices of order n i.e A' = A and B' = B

⇒ (A + B)' = A + B

Hence, statement 1 is true.

Statement 2: A ⋅ B is a symmetric matrix

Let's find out the transpose of A ⋅ B

⇒ (A ⋅ B)' = B' ⋅ A'

∵ A and B are two symmetric matrices of order n i.e A' = A and B' = B

⇒ (A ⋅ B)' = B ⋅ A

But we also know that matrix multiplication is not commutative in general

So, we cannot say that (A ⋅ B)' = A ⋅ B in general

Hence, statement 2 is false.

Concept:

• Symmetric Matrix: Any real square matrix A = (aij) is said to be a symmetric matrix if and only if aij = aji, ∀ i, and j in other words we can say that if A is a real square matrix such that A = A’ then A is said to be a symmetric matrix.
• Skew-symmetric Matrix: Any real square matrix A = (aij) is said to be a skew-symmetric matrix if and only if aij = - aji, ∀ I, and j or in other words we can say that if A is a real square matrix such that A =- A’ then A is said to be a skew-symmetric matrix.

• Any real square matrix says A, can be expressed as the sum of the symmetric and skew-symmetric matrix.

  i.e  where A + A’ is symmetric and A – A’ is a skew-symmetric matrix.

Calculation:

Given:  

where P is symmetric and Q is a skew-symmetric matrix

Here we have to find the matrix P and Q

As we know, any square matrix can be expressed as the sum of the symmetric and skew-symmetric matrices.

i.e If A is a square matrix then A can be expressed as

where A + A’ is symmetric and A – A’ is a skew-symmetric matrix.

By comparing  with  we get,

⇒ P = A + A' and Q = A - A'

Similarly,

Hence, 

Concept:  

A symmetric matrix 

• It is a square matrix A of size n × n when the square matrix is equal to the transposed form of that matrix, that is, A^T = A.
• If A = [a_ij]_n×n is the symmetric matrix, then a_ij = a_ji for 1 ≤ i ≤ n, and 1 ≤ j ≤ n.

A skew-symmetric matrix 

• It is a square matrix A of size n × n when the square matrix is equal to the transposed form of that matrix, that is, A^T = -A.
• Diagonal elements of the skew-symmetric is zero.

If A = [a_ij]_n×n is the skew-symmetric matrix, then a_ij = -a_ji for 1 ≤ i ≤ n, and 1 ≤ j ≤ n.

A square matrix is called a singular matrix when its determinant is equal to 0.

A square matrix is called a non-singular matrix when its determinant is not equal to 0.

Calculation:

Given:

The given matrix is A = .

Transpose of the matrix is A^t = .

Since A^t ≠ A, hence the matrix A is not symmetric.

Since diagonal elements of the matrix are not zero, then matrix A is not skew-symmetric.

The determinant ∆ of the matrix is given by,

∆ = 1(1(1) - 2(-2)) - (-2)(1(2) - 3(-2)) + (-3)(2(2) - 1(3))

 ∆ = 1(1 + 4) + 2(2 + 6) - 3(4 - 3)

 ∆ = 5 + 16 - 3

 ∆ = 18

The determinant of the matrix is not equal to zero, hence it is non-singular.

Hence, the correct answer is option 4.