Matrices MCQ Quiz - Objective Question with Answer for Matrices - Download Free PDF

Last updated on Jun 14, 2025

Latest Matrices MCQ Objective Questions

Matrices Question 1:

If 

where x,y,z are integers, is an orthogonal matrix, then what is A2 equal to?

  1. Null matrix 
  2. Identity matrix
  3. A
  4. -A

Answer (Detailed Solution Below)

Option 2 : Identity matrix

Matrices Question 1 Detailed Solution

Calculation:

Given,

Matrix A is defined as:

It is mentioned that A is an orthogonal matrix. Therefore,

Now, we calculate A2:

Since A is orthogonal, , and hence:

∴ A2 = Identity matrix (I).

Hence, the Correct answer is Option 2.

Matrices Question 2:

If  then what is A24A equal to?

  1. 5I3
  2. I3
  3. I3
  4. 5I3

Answer (Detailed Solution Below)

Option 4 : 5I3

Matrices Question 2 Detailed Solution

Calculation:

Given,

 

Now 4A

Also A2 - 4A

Relate the result to the identity matrix I3" id="MathJax-Element-594-Frame" role="presentation" style="position: relative;" tabindex="0">I3I3" id="MathJax-Element-39-Frame" role="presentation" style="position: relative;" tabindex="0">I3I3

Hence, the correct answer is option 4.

Matrices Question 3:

If  then what is (f(π))2 equal to?

Answer (Detailed Solution Below)

Option 4 :

Matrices Question 3 Detailed Solution

Concept:

Rotation Matrix:

  • A rotation matrix is used to perform a rotation in a Euclidean space. It is a square matrix that describes the rotation of a vector space.
  • For a 2D rotation, the matrix is given by:
  • Here, θ is the angle of rotation in radians.
    • cos θ: Represents the cosine of the rotation angle.
    • sin θ: Represents the sine of the rotation angle.
  • Key property of a rotation matrix:
    • The transpose of the matrix is equal to its inverse.
    • The determinant of the matrix is always equal to 1.
  • When θ = π, the rotation matrix becomes:

 

Calculation:

Given,

Rotation matrix at θ = π:

To find (f(π))2, multiply the matrix by itself:

Using matrix multiplication:

Top-left element: (-1)(-1) + (0)(0) = 1

Top-right element: (-1)(0) + (0)(-1) = 0

Bottom-left element: (0)(-1) + (-1)(0) = 0

Bottom-right element: (0)(0) + (-1)(-1) = 1

Resulting matrix:

∴ (f(π))2 is equal to the identity matrix, which is

Hence, the correct answer is Option 4.

Matrices Question 4:

If

 

where x,y,z are integers, is an orthogonal matrix, then what is the value of x2+y2+z2?

  1. 0
  2. 1
  3. 4
  4. 14

Answer (Detailed Solution Below)

Option 2 : 1

Matrices Question 4 Detailed Solution

Calculation:

Given,

The matrix A is:

Since A  is an orthogonal matrix, we know that:

This property tells us that A  is orthogonal, and it implies that (the product of A's transpose and A is equal to the identity matrix I , which is:

Now, let’s calculate step by step. The transpose of matrix A , denoted  is:

Now, we perform matrix multiplication between and A:

Performing this multiplication, we get the following matrix:

This matrix must be equal to the identity matrix I , which is:

By comparing the elements of the matrices, we get the following system of equations:

1. 2.  3.

Thus, the key result from the orthogonality condition is:

Hence, the correct answer is Option 2. 

Matrices Question 5:

If

 

then which one of the following is a value of x?

  1. -2
  2. -1
  3. 0
  4. 1

Answer (Detailed Solution Below)

Option 4 : 1

Matrices Question 5 Detailed Solution

Calculation:

Multiply Matrix 1 and Matrix 2:

 

Multiply the resulting matrix with Matrix 3:

Equate the result to 45:

 

 

Step 5: Verify:

For , substitute back:

45 =45

∴ The correct value of x is 1.

Hence, the correct answer is Option 4.

Top Matrices MCQ Objective Questions

If A =  is a symmetric matrix then x

  1. 3
  2. 6
  3. 8
  4. 0

Answer (Detailed Solution Below)

Option 2 : 6

Matrices Question 6 Detailed Solution

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Concept:

Symmetric Matrix:

  • Square matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself
  • AT = A or A’ = A

Where, AT or A’ denotes the transpose of matrix

  • A square matrix A is said to be symmetric if aij = aji for all i and j

Where aij and aji is an element present in matrix.

 

Calculation:

Given:

A is a symmetric matrix,

⇒ AT = A or aij = aji

A =

So, by property of symmetric matrices

⇒ a12 = a21

⇒ x – 3 = 3

∴ x = 6

lf the order of A is 4 × 3, the order of B is 4 × 5 and the order of C is 7 × 3, then the order of (ATB)T C T is

  1. 5 × 3
  2. 4 × 5
  3. 5 × 7
  4. 4 × 3

Answer (Detailed Solution Below)

Option 3 : 5 × 7

Matrices Question 7 Detailed Solution

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Concept:

  • To multiply an m × n matrix by an n × p matrix, the n must be the same, and the result is an m × p matrix.
  • If A be a matrix of order m × n than the order of transpose matrix is n × m

Calculation:

Given:

Order of A is 4 × 3, the order of B is 4 × 5 and the order of C is 7 × 3

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

So, order of AT is 3 × 4 and order of CT is 3 × 7

Now,

ATB = {3 × 4} {4 × 5} = 3 × 5

⇒ Order of ATB is 3 × 5

Hence order of (ATB) T is 5 × 3

Now order of (ATB) T C T = {5 × 3} {3 × 7} = 5 × 7

∴ Order of (ATB) T C T is 5 × 7

If  is symmetric, then what is x equal to?

  1. 2
  2. 3
  3. -1
  4. 5

Answer (Detailed Solution Below)

Option 4 : 5

Matrices Question 8 Detailed Solution

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Concept:

Symmetric Matrix: If the transpose of a matrix is equal to itself, that matrix is said to be symmetric.

Or, A matrix A is symmetric if and only if swapping indices doesn't change its components

  • A = AT
  • aij = aji

 

CALCULATION:

Given -

A real square matrix A = (aij) is said to be symmetric, if A = AT

Where AT = transpose of matrix A

∴ A = AT

Compare A21 element.

⇒ x + 2 =2x - 3 

⇒ x = 5

If , then k = ?

  1. - 25
  2. - 15
  3. None of these.

Answer (Detailed Solution Below)

Option 3 :

Matrices Question 9 Detailed Solution

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Concept:

For an invertible matrix A:

  • A-1.
  • |A-1| = |A|-1.

 

Calculation:

         -----(1)

From the definition of the inverse of a matrix, 

A-1 =               -----(2)

Comparing equation (1) & (2), we get

k = |A|  

Using the properties of the determinant of inverse of a matrix, we have:

k = |A| =         ----(3)

We know, 

A.A-1 = I

⇒ |A.A-1| = |I| = 1

⇒ |A| |A-1| = 1

⇒ |A| = 1/ |A-1|       ....(4)

Now,

|A-1| = 1(24 - 3) + 2(9 - 12) + 3(2 - 12) = 21 - 6 - 30 = - 15.

|A-1| = -15

Therefore, from equation (3)

k = .

Mistake PointsNote that, we have A-1 matrix, not an A matrix. So to find the value of k, don't you have to use relation |A| = 1/|A-1|

If  and \(\rm A ^{-1}=\begin{bmatrix} {1\over8} & {-1\over 12} \\\ {-1\over 6}& {4\over 9} \end{bmatrix}\), then find the value of x?

  1. 10

Answer (Detailed Solution Below)

Option 2 :

Matrices Question 10 Detailed Solution

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Concept:

A × A-1 = I, where I is an identity matrix

|A| = 

Calculation:

Given:  and \(\rm A ^{-1}=\begin{bmatrix} {1\over8} & {-1\over 12} \\\ {-1\over 6}& {4\over 9} \end{bmatrix}\)

|A-1| = 

|A| =  = 24

⇒ 3x - 8 = 24

∴ x = 

If , then trace of matrix A is

  1. 6
  2. 7
  3. 8
  4. 9

Answer (Detailed Solution Below)

Option 2 : 7

Matrices Question 11 Detailed Solution

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Concept:

Trace of a matrix:

Trace of a matrix is the sum of elements on the main diagonal.

The trace is only defined for a square matrix (n × n).

Let A be n × n matrix. 

Calculation:

Given: 

Trace of matrix = sum of elements on the main diagonal

= -1 + 8

= 7

Consider the following question and decide which of the statements is sufficient to answer the question.

Find the value of n, if

Statements∶

1. AB = A

2. 

  1. Only 1 is sufficient
  2. Only 2 is sufficient
  3. Either 1 or 2 is sufficient
  4. Both 1 and 2 are not sufficient

Answer (Detailed Solution Below)

Option 4 : Both 1 and 2 are not sufficient

Matrices Question 12 Detailed Solution

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Concept:

Multiplication of matrices:

  • The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix.
  • The result will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.
  • To multiply an m × n matrix by an n × p matrix, the n must be the same, and the result is an m × p matrix.

Calculation:

From statement 1

AB = A

We cannot find anything from this statement.

From statement 2

We cannot find anything from this statement.

Combining statement 1 and 2

Also, 

∴ We cannot find the value of n from both statements together.

Each entry is the number of all possible matrices of 3 × 3 with 0 or 1, respectively.

  1. 9
  2. 18
  3. 27
  4. 512

Answer (Detailed Solution Below)

Option 4 : 512

Matrices Question 13 Detailed Solution

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Calculation:

As we know that

Number of possible entries of 3 × 3 = 9

And, every entry has two choices = 0 and 1

Now,

Total number of choices = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

⇒ 29

⇒ 512

∴ The total number of choices is 512.

If A is a square matrix such that A2 = I, then A3 + (A + I)2 - 9A - I- A2 is

  1. -10A
  2. 10A
  3. -6A
  4. 6A

Answer (Detailed Solution Below)

Option 3 : -6A

Matrices Question 14 Detailed Solution

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Concept:

Properties of identity matrix:

If A is the square matrix of order n × n

  • AI = IA = A
  • In = I           (Where n ∈ N)

 

Calculation:

Given

A2 = I

Now, A3 + (A + I)2 - 9A - I2 - A2

= A2. A + A2 + I2 + 2AI - 9A - I- A2

= I. A + I + I + 2AI - 9A - I - I           [∵ A2 = I and AI = IA = A]

= AI + 2AI - 9A 

= 3AI - 9A

= 3A - 9A

= - 6A

If A is an Involuntary matrix and I is a unit matrix of same order, then (I − A) (I + A) is

  1. A
  2. I
  3. 2A
  4. Zero matrix

Answer (Detailed Solution Below)

Option 4 : Zero matrix

Matrices Question 15 Detailed Solution

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Concept:

Involuntary matrix:

  • Matrix A is said to be Involuntary if A2 = I, where I is an Identity matrix of same order as of A.
  • Involuntary matrix is a matrix that is equal to its own inverse. ⇔ A-1 = A

 

Calculation:

Given that A is involuntary matrix,

⇒ A2 = I

Now,

(I − A) (I + A) = I2 – IA + AI − A2 

⇒ I – A + A – I         (∵ A2 = I)

0

∴ (I − A) (I + A) is zero matrix.

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