Matrices & Determinants for Competitive Exam Success

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Matrices & Determinants

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Matrices & Determinants MCQ & Objective Questions

Matrices and determinants are crucial topics in mathematics that play a significant role in various examinations. Mastering these concepts not only enhances your problem-solving skills but also boosts your confidence in tackling objective questions. Practicing MCQs and important questions related to matrices and determinants is essential for effective exam preparation, helping you score better in your assessments.

What You Will Practise Here

Understanding the definition and types of matrices
Matrix operations: addition, subtraction, and multiplication
Determinants: calculation methods and properties
Inverse of a matrix and its applications
Rank of a matrix and its significance
Solving linear equations using matrices
Applications of matrices in real-world problems

Exam Relevance

The topic of matrices and determinants is frequently included in the syllabi of CBSE, State Boards, NEET, and JEE. Students can expect questions that test their understanding of matrix operations, determinant calculations, and their applications in solving equations. Common question patterns often involve direct computation, conceptual understanding, and application-based problems, making it vital to be well-prepared.

Common Mistakes Students Make

Confusing the properties of determinants with those of matrices
Errors in calculating determinants, especially for larger matrices
Neglecting the importance of matrix dimensions in operations
Misunderstanding the concept of the inverse of a matrix
Overlooking the application of matrices in solving real-life problems

FAQs

Question: What are the types of matrices I should know for exams?
Answer: You should be familiar with row matrices, column matrices, square matrices, and zero matrices, as they are commonly tested.

Question: How can I calculate the determinant of a 3x3 matrix?
Answer: You can use the rule of Sarrus or the cofactor expansion method to find the determinant of a 3x3 matrix.

Question: Why is it important to practice MCQs on matrices and determinants?
Answer: Practicing MCQs helps reinforce your understanding, improves speed, and prepares you for the types of questions you will encounter in exams.

Get ready to enhance your skills! Dive into our practice MCQs on matrices and determinants to test your understanding and excel in your exams.

Determinants Inverse of Matrix Matrix Operations Properties Types of Matrices

Q. Calculate the determinant of the matrix J = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

A. 0
B. 1
C. 2
D. 3

Solution

The determinant of J is calculated as 1*(1*1 - 0*3) - 2*(0*1 - 0*2) + 1*(0*3 - 1*2) = 1 - 0 - 2 = -1.

Correct Answer: B — 1

Q. Calculate the determinant of the matrix \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2023)

A. 10
B. 11
C. 12
D. 13

Solution

Det(I) = (3*4) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

Q. Find the determinant of the matrix \( D = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \). (2019)

A. -2
B. 2
C. 4
D. 0

Solution

Det(D) = (1*4) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

Q. Find the eigenvalues of the matrix G = [[2, 1], [1, 2]]. (2020)

A. 1, 3
B. 2, 2
C. 3, 1
D. 0, 4

Solution

The characteristic polynomial is det(G - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0. The eigenvalues are λ = 1 and λ = 3.

Correct Answer: A — 1, 3

Q. Find the eigenvalues of the matrix G = [[5, 4], [2, 3]]. (2020)

A. 1, 7
B. 2, 6
C. 3, 5
D. 4, 4

Solution

The eigenvalues are found by solving the characteristic equation det(G - λI) = 0. This gives λ^2 - 8λ + 7 = 0, which factors to (λ - 1)(λ - 7) = 0, hence λ = 1, 7.

Correct Answer: A — 1, 7

Q. Find the inverse of the matrix D = [[4, 7], [2, 6]]. (2023)

A. [[3/2, -7/4], [-1/2, 2/4]]
B. [[3/2, -7/4], [-1/4, 2/4]]
C. [[6, -7], [-2, 4]]
D. [[6, 7], [2, 4]]

Solution

The inverse of D is (1/det(D)) * adj(D) = (1/(4*6 - 7*2)) * [[6, -7], [-2, 4]] = [[3/2, -7/4], [-1/2, 2/4]].

Correct Answer: A — [[3/2, -7/4], [-1/2, 2/4]]

Q. Find the inverse of the matrix F = [[4, 7], [2, 6]]. (2021)

A. [[3, -7], [-1, 4]]
B. [[6, -7], [-2, 4]]
C. [[3, 7], [-1, 2]]
D. [[2, -7], [-1, 4]]

Solution

The inverse of F is given by (1/det(F)) * adj(F). Here, det(F) = 4*6 - 7*2 = 10, and adj(F) = [[6, -7], [-2, 4]]. Thus, F^(-1) = (1/10) * [[6, -7], [-2, 4]] = [[3/5, -7/10], [-1/5, 2/5]].

Correct Answer: A — [[3, -7], [-1, 4]]

Q. For the matrix D = [[4, 2], [1, 3]], find the inverse of D. (2022)

A. [[3, -2], [-1, 4]]
B. [[3, 2], [-1, 4]]
C. [[3, -2], [1, 4]]
D. [[4, -2], [-1, 3]]

Solution

The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10. The adjugate is [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].

Correct Answer: A — [[3, -2], [-1, 4]]

Q. For the matrix J = [[0, 1], [1, 0]], what is J^2?

A. [[1, 0], [0, 1]]
B. [[0, 1], [1, 0]]
C. [[0, 0], [0, 0]]
D. [[1, 1], [1, 1]]

Solution

Calculating J^2 gives [[0, 1], [1, 0]] * [[0, 1], [1, 0]] = [[1, 0], [0, 1]], which is the identity matrix.

Correct Answer: A — [[1, 0], [0, 1]]

Q. For the matrix J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (2023)

A. -24
B. 24
C. 0
D. 12

Solution

Using the determinant formula, det(J) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) + 3*(-5) = -24 + 40 - 15 = 1.

Correct Answer: A — -24

Q. For the matrix \( F = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \), what is the value of the determinant? (2021)

A. 5
B. 6
C. 7
D. 8

Solution

Det(F) = (2*4) - (1*3) = 8 - 3 = 5.

Correct Answer: A — 5

Q. If A = [[2, 3], [1, 4]], what is A^2? (2020)

A. [[7, 18], [18, 7]]
B. [[12, 21], [21, 12]]
C. [[12, 21], [21, 16]]
D. [[10, 21], [21, 10]]

Solution

A^2 = A * A = [[2*2 + 3*1, 2*3 + 3*4], [1*2 + 4*1, 1*3 + 4*4]] = [[10, 21], [21, 16]].

Correct Answer: C — [[12, 21], [21, 16]]

Q. If A = [[2, 3], [1, 4]], what is the inverse of A?

A. [[4, -3], [-1, 2]]
B. [[4, 3], [-1, 2]]
C. [[2, -3], [-1, 2]]
D. [[3, -2], [-1, 2]]

Solution

The inverse of A is given by (1/det(A)) * adj(A). Det(A) = (2*4) - (3*1) = 5. The adjoint is [[4, -3], [-1, 2]]. Thus, A^(-1) = (1/5) * [[4, -3], [-1, 2]].

Correct Answer: A — [[4, -3], [-1, 2]]

Q. If A is a 2x2 matrix and B is a 2x2 matrix, what is the order of the product AB? (2019)

A. 2x2
B. 2x3
C. 3x2
D. 3x3

Solution

The order of the product of two matrices is determined by the outer dimensions. Since both A and B are 2x2 matrices, their product AB will also be a 2x2 matrix.

Correct Answer: A — 2x2

Q. If A is a 2x2 matrix and B is a 2x3 matrix, what is the order of the product AB? (2019)

A. 2x2
B. 2x3
C. 3x2
D. 2x5

Solution

The order of the product of two matrices is determined by the outer dimensions. Here, A (2x2) and B (2x3) can be multiplied, resulting in a matrix of order 2x3.

Correct Answer: B — 2x3

Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum number of non-zero elements in A + B? (2021)

A. 9
B. 6
C. 3
D. 0

Solution

The maximum number of non-zero elements in the sum of two matrices occurs when all elements of both matrices are non-zero. Therefore, A + B can have a maximum of 9 non-zero elements.

Correct Answer: A — 9

Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum order of the resultant matrix when A is multiplied by B? (2022)

A. 3x3
B. 6x6
C. 9x9
D. 3x6

Solution

The order of the resultant matrix when two matrices are multiplied is determined by the outer dimensions. Here, both A and B are 3x3, so the product AB is also 3x3.

Correct Answer: A — 3x3

Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum order of the resultant matrix when A is added to B? (2021)

A. 3x3
B. 3x2
C. 2x3
D. 3x1

Solution

The sum of two matrices of the same order results in a matrix of the same order. Therefore, A + B will be a 3x3 matrix.

Correct Answer: A — 3x3

Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum order of the resultant matrix AB? (2023)

A. 3x3
B. 2x2
C. 3x2
D. 2x3

Solution

The product of two matrices A and B, both of order 3x3, will also be a matrix of order 3x3.

Correct Answer: A — 3x3

Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the order of A + B? (2023)

A. 3x3
B. 3x2
C. 2x3
D. 3x1

Solution

The order of the sum of two matrices is the same as the order of the individual matrices. Therefore, A + B is a 3x3 matrix.

Correct Answer: A — 3x3

Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the order of the matrix A + B? (2019)

A. 3x3
B. 3x2
C. 2x3
D. 3x1

Solution

The sum of two matrices is defined only when they have the same order. Since both A and B are 3x3 matrices, A + B will also be a 3x3 matrix.

Correct Answer: A — 3x3

Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the order of the matrix product AB? (2021)

A. 3x3
B. 2x2
C. 3x2
D. 2x3

Solution

The product of two matrices A and B, both of order 3x3, will also be a matrix of order 3x3.

Correct Answer: A — 3x3

Q. If A is a 3x3 matrix, how many elements does it have? (2023)

A. 6
B. 9
C. 12
D. 3

Solution

A 3x3 matrix has 3 rows and 3 columns, resulting in a total of 3 * 3 = 9 elements.

Correct Answer: B — 9

Q. If A is a 3x3 matrix, what is the maximum number of linearly independent rows it can have? (2023)

A. 1
B. 2
C. 3
D. 4

Solution

A 3x3 matrix can have a maximum of 3 linearly independent rows, but the answer options are incorrect. The correct answer is 3.

Correct Answer: C — 3

Q. If a matrix has more columns than rows, it is called a: (2022)

A. Row matrix
B. Column matrix
C. Rectangular matrix
D. Square matrix

Solution

A matrix with more columns than rows is referred to as a rectangular matrix.

Correct Answer: C — Rectangular matrix

Q. If a matrix is both upper triangular and lower triangular, what type of matrix is it? (2020)

A. Zero matrix
B. Identity matrix
C. Diagonal matrix
D. Square matrix

Solution

A matrix that is both upper and lower triangular must have non-zero elements only on the diagonal, making it a diagonal matrix.

Correct Answer: C — Diagonal matrix

Q. If a matrix is diagonal, what can be said about its non-diagonal elements? (2020)

A. They are all zero
B. They are all one
C. They can be any value
D. They are negative

Solution

In a diagonal matrix, all non-diagonal elements are zero.

Correct Answer: A — They are all zero

Q. If a matrix is diagonal, which of the following must be true? (2020)

A. All elements are zero
B. Only diagonal elements are non-zero
C. All elements are equal
D. It is a square matrix

Solution

A diagonal matrix has non-zero elements only on its main diagonal, while all other elements are zero.

Correct Answer: B — Only diagonal elements are non-zero

Q. If a matrix is said to be orthogonal, what property does it have?

A. All elements are zero
B. Transpose is equal to its inverse
C. All diagonal elements are equal
D. It is a square matrix

Solution

An orthogonal matrix is defined as a square matrix whose transpose is equal to its inverse.

Correct Answer: B — Transpose is equal to its inverse

Q. If a matrix is said to be skew-symmetric, what must be true about its elements? (2023)

A. All elements are zero
B. a_ij = -a_ji
C. a_ij = a_ji
D. All diagonal elements are zero

Solution

A skew-symmetric matrix satisfies the condition a_ij = -a_ji for all i and j.

Correct Answer: B — a_ij = -a_ji

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