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?
Answer (Detailed Solution Below)
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,
∴ A2 = Identity matrix (I).
Hence, the Correct answer is Option 2.Matrices Question 2:
If
Answer (Detailed Solution Below)
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">
Hence, the correct answer is option 4.
Matrices Question 3:
If
Answer (Detailed Solution Below)
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
Matrices Question 4:
If
where x,y,z are integers, is an orthogonal matrix, then what is the value of x2+y2+z2?
Answer (Detailed Solution Below)
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
Now, let’s calculate
Now, we perform matrix multiplication between
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.
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?
Answer (Detailed Solution Below)
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
45 =45
∴ The correct value of x is 1.
Hence, the correct answer is Option 4.
Top Matrices MCQ Objective Questions
If A =
Answer (Detailed Solution Below)
Matrices Question 6 Detailed Solution
Download Solution PDFConcept:
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
Answer (Detailed Solution Below)
Matrices Question 7 Detailed Solution
Download Solution PDFConcept:
- 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
Answer (Detailed Solution Below)
Matrices Question 8 Detailed Solution
Download Solution PDFConcept:
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
Answer (Detailed Solution Below)
Matrices Question 9 Detailed Solution
Download Solution PDFConcept:
For an invertible matrix A:
- A-1 =
. - |A-1| = |A|-1 =
.
Calculation:
From the definition of the inverse of a matrix,
A-1 =
Comparing equation (1) & (2), we get
k = |A|
Using the properties of the determinant of inverse of a matrix, we have:
k = |A| =
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
Answer (Detailed Solution Below)
Matrices Question 10 Detailed Solution
Download Solution PDFConcept:
A × A-1 = I, where I is an identity matrix
|A| =
Calculation:
Given:
|A-1| =
|A| =
⇒ 3x - 8 = 24
∴ x =
If
Answer (Detailed Solution Below)
Matrices Question 11 Detailed Solution
Download Solution PDFConcept:
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.
Answer (Detailed Solution Below)
Matrices Question 12 Detailed Solution
Download Solution PDFConcept:
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.
Answer (Detailed Solution Below)
Matrices Question 13 Detailed Solution
Download Solution PDFCalculation:
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 - I2 - A2 is
Answer (Detailed Solution Below)
Matrices Question 14 Detailed Solution
Download Solution PDFConcept:
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 - I2 - 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
Answer (Detailed Solution Below)
Matrices Question 15 Detailed Solution
Download Solution PDFConcept:
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.