Transpose of a Matrix MCQ Quiz - Objective Question with Answer for Transpose of a Matrix - Download Free PDF

Calculation

Given: and

Hence option 2 is correct.

Concept

(A + B)^t = A^t + B^t

Calculation

Given:

 .....(1)

 .......(2)

Transpose (1): .....(3)

Add (2) and (3):

⇒ 

⇒ 

⇒ 

Hence option 4 is correct

Concept -

If A =  then transpose(A) = A' = 

Explanation -

We have  A' =  and B =  , here A' is the transpose of A.

Now A = 

So 3A =  & 2B = 

Now 3A + 2B =  +  = 

Now (3A + 2B)' = 

Hence Option (3) is correct.

Concept -

If A =  then transpose(A) = A' = 

Explanation -

We have  A' =  and B =  , here A' is the transpose of A.

Now A = 

So 3A =  & 2B = 

Now 3A + 2B =  +  = 

Now (3A + 2B)' = 

Hence Option (3) is correct.

Given:

A is a 2 × 2 matrix such that

1. A + AT = I
2. ATA = I​

Concept:

• Matrix multiplication is row into column-wise.
• Transpose of a matrix A represented by AT is obtained by interchanging rows into columns and vice-versa.

​Solution:

Let A = 

then AT = 

∵ A + AT = I

⇒  = 

On equating the elements -

2a = 1

⇒ a = 1/2    - (i)

b + c = 0

⇒ b = -c    - (ii)

2d = 1

⇒ d = 1/2    - (iii)

Also, 

ATA = I

⇒  = 

Putting values of a, b, c from (i), (ii) and (iii) -

We see a2 + c2 = 1

⇒ c2 = 3/4

⇒ c = ± 

Similarly, we can see, b = ± 

So, possible matrices are  and 

So, the total number of matrices = 2

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

It is denoted by A′ or AT

Calculation:

Given A =  and B = 

AB =  × 

AB = 

AB = 

∴ (AB)^T = 

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called the transpose of the matrix.

It is denoted by A′ or A^T

Equal matrix:

Two matrices are equal if they have the same dimension or order and the corresponding elements are identical.

Calculation:

Given and A + AT = I

A + A^T =  +  = I

= 

∴ 2cos 2θ = 1

cos 2θ = 

2θ = 

θ = 

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

For example: .

It is denoted by A' or A^T.

Properties of Transpose of a Matrix:

• The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order:

  (AB)' = B'A'

• (ABC)' = C'B'A'

• (A')' = A

Calculation:

It is given that B is a skew-symmetric matrix.

∴ B' = -B

Now, consider the transpose of the product matrix A′BA.

(A′BA)' = A'B'(A')'

= A'(-B)A               [∵ B' = -B]

= -(A'BA)

Since the transpose is equal to its negative, A'BA is a Skew-Symmetric matrix.

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

It is denoted by A′ or A^T

Calculation:

Given:

A = 

Now find the transpose of a matrix A,

A^T = 

Now,

A - AT = 

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called the transpose of the matrix.

It is denoted by A′ or AT

Calculation:

Given:

A = 

Now find the transpose of a matrix A,

AT = 

Now,

A + AT = 

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

For example: 

It is denoted by  or A^T.

Calculations:

Given, 

A' can be obtained by interchanging row and column of matrix A

Consider, AA' =

⇒AA' = 

⇒AA' = 

⇒AA' = I

Hence, If , then AA' = I

Concept:

1). Transpose of a Matrix:

     The new matrix obtained by interchanging the rows and columns of the original matrix is called

     the transpose of the matrix.

     For example: ⇒ 

     It is denoted by A' or A^T.
2). For the addition and subtraction of two matrices, the order of matrices should be equal.

Calculation:

Given:

and 

C = A + B

⇒ 

⇒ 

⇒ ∣C∣ = 2(4 - 0)

⇒ ∣C∣ = 8

∴ The determinant of matrix C is 8.

Concept:

Transpose of a sum: The transpose of the sum of two matrices is equivalent to the sum of their transposes: (A + B)^T = A^T + B^T

Transpose of a product: The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order: (AB)^T = B^T A^T

Calculation:

Given: (AB + B)^T = B^TX

⇒ (AB + B)^T 

= (AB)^T + B^T

= B^TA^T + B^TI

= B^T (A^T + I) = B^T X

∴ X = (A^T + I)

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

For example: 

• It is denoted by  or A^T.

Properties of Transpose of a Matrix:

• The transpose of the transpose of a matrix is the matrix itself: .
• The transposes of equal matrices are also equal: .
• The transpose of the sum/difference of two matrices is equivalent to the sum/difference of their transposes: .
• The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order: .

Calculation: 

Using the Properties of Transpose of a Matrix:

Also, 

Adding equations (1) and (2), we get,

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

It is denoted by  or AT.

For example: 

Calculation:

Let matrix A =

a₁₁ = 2 × (1) - 1 = 1

a₁₂ = 2 × (1) - 2 = 0

a₁₃ = 2 × (1) - 3 = -1

a₂₁ = 2 × (2) - 1 = 3

a₂₂ = 2 × (2) - 2 = 2

a₂₃ = 2 × (2) - 3 = 1

a₃₁ = 2 × (3) - 1 = 5

a₃₂ = 2 × (3) - 2 = 4

a₃₃ = 2 × (3) - 3 = 3

∴ A = 

A^T = 

⇒ A^T = 

Additional Information

Properties of Transpose of a Matrix:

• The transpose of a matrix is the matrix itself:
• The transposes of equal matrices are also similar:
• The transpose of the sum/difference of two matrices is equivalent to the sum/difference of their transposes:
• The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order: