Tough Matrices And Determinants Test - 82+ MCQs: Matrices and Determinants Quiz: Test Your Knowledge with Matrix Algebra Questions

1. How do you find the inverse of a 2x2 matrix?

Adjoint divided by determinant

Transpose divided by determinant

Determinant divided by adjoint

Determinant divided by transpose

2. In matrix calculus, what is the derivative of the determinant of a matrix with respect to the matrix itself?

The inverse of the matrix

The adjugate of the matrix

The transpose of the matrix

The cofactor matrix

3. If a matrix has only one row, what is it called?

Scalar matrix

Row matrix

Column matrix

Square matrix

4. In Gaussian elimination, what is the goal?

To reduce a matrix to echelon form

To find the determinant

To calculate the rank

To perform matrix multiplication

5. When is a matrix diagonalizable?

When it has only one row

When its determinant is zero

When it is invertible

When it has enough linearly independent eigenvectors

6. What is the Matrix Sign Function used for?

Computing the square root of a matrix

Computing the determinant of a matrix

Computing the inverse of a matrix

Computing the sign of the eigenvalues

7. What is the sum of the elements in the main diagonal of a square matrix called?

Determinant

Trace

Eigenvalue

Transpose

8. What is the result of multiplying a matrix by the identity matrix?

The original matrix

A matrix of zeros

A matrix with all ones

The identity matrix

9. What is the Sylvester equation in matrix algebra?

A determinant equation

A system of linear equations

An eigenvalue equation

A matrix equation involving two matrices

10. What is the adjugate of a matrix?

The transpose of its cofactor matrix

The inverse of the matrix

The determinant of the matrix

The product of its diagonals

11. In a 3x3 matrix, what is the cofactor of an element at position (i, j)?

The determinant of the matrix excluding the i-th row and j-th column

The product of the i-th row and j-th column

The reciprocal of the element at position (i, j)

The sum of the elements in the i-th row and j-th column

12. What is the purpose of the cross product in vector operations?

Scalar multiplication

Dot product

Orthogonal projection

Vector addition

13. What is the spectral radius of a matrix?

The maximum absolute eigenvalue

The sum of eigenvalues

The determinant of the matrix

The product of eigenvalues

14. What is the transpose of a matrix?

Flipping the matrix over its diagonal

Swapping rows and columns

Negating all elements

Doubling all elements

15. In a 2x2 matrix, what is the formula for finding the inverse?

1/Det(A) * adj(A)

1/Det(A) * Transpose(A)

1/Det(A) * inverse(A)

1/Det(A) * cofactor(A)

16. What does the determinant of a matrix represent?

The area of the matrix

The volume of the matrix

The scaling factor of the matrix

The sum of matrix elements

17. What is the rank of a matrix?

The sum of its elements

The number of non-zero rows in its echelon form

The determinant of the matrix

The product of its elements

18. What is the sum of a matrix and its additive inverse?

The identity matrix

A zero matrix

A matrix of ones

The original matrix

19. In matrix terminology, what does the term 'singular' refer to?

A matrix with only one element

A matrix with all zero elements

A non-square matrix

A matrix with no inverse

20. In a 3x3 matrix, how many elements are there?

21. What happens to the determinant of a matrix when two rows are exchanged?

It remains the same

It becomes zero

It is squared

It is halved

22. Which operation does not change the determinant of a matrix?

Multiplying a row by a constant

Swapping two rows

Adding a multiple of one row to another

Taking the transpose

23. What is the Hessenberg form of a matrix?

A factorization into a diagonal matrix and an orthogonal matrix

A factorization into a symmetric matrix and an antisymmetric matrix

A factorization into an upper triangular matrix and a lower triangular matrix

A factorization into a block diagonal matrix and a block off-diagonal matrix

24. In the context of linear algebra, what is a Grassmannian?

A set of orthogonal matrices

A set of subspaces

A set of eigenvalues

A set of rank-deficient matrices

25. Which matrix is the result of subtracting matrix B from matrix A?

A + B

B - A

A * B

A / B

26. Given matrices A and B, what is the result of the matrix product A * B?

Matrix C such that c_ij = Σ(A_ik * B_kj) for all k

Matrix C such that c_ij = A_ij * B_ij

Matrix C such that c_ij = A_ij + B_ij

Matrix C such that c_ij = A_ij - B_ij

27. Which property does the inverse of a matrix satisfy?

Commutativity

Associativity

Distributivity

Cancellation

28. What is the Schur decomposition of a matrix?

A factorization into a unitary matrix and an upper triangular matrix

A factorization into a diagonal matrix and an orthogonal matrix

A factorization into a symmetric matrix and an antisymmetric matrix

A factorization into a block diagonal matrix and a block off-diagonal matrix

29. What is the determinant of an identity matrix?

Infinity

Undefined

30. In matrix multiplication, what is the number of columns in the first matrix required to be equal to?

Number of rows in the second matrix

Number of columns in the second matrix

Number of elements in the second matrix

Number of rows in the first matrix

31. What is the inverse of a square matrix A denoted as?

A^2

1/A

A^-1

32. Which property does the cross product of two vectors satisfy?

Associativity

Commutativity

Distributivity

Anticommutativity

33. When is a matrix considered defective?

When it has a repeated eigenvalue

When it has no eigenvalues

When it is a square matrix

When it is invertible

34. In Cramer's Rule, what does each determinant represent?

Coefficients of the variables

Solution of the system of equations

Inverse of the matrix

Scaling factor of the matrix

35. What is the singular value decomposition (SVD) of a matrix?

A factorization of the matrix into three simpler matrices

The determinant divided by the trace

The inverse of the matrix

A diagonal matrix with singular values on the main diagonal

36. What is the Cayley-Hamilton theorem?

Every matrix is diagonalizable

Every square matrix satisfies its own characteristic equation

The determinant of a matrix is equal to the sum of its eigenvalues

The trace of a matrix is equal to the sum of its eigenvalues

37. What is the significance of the null space of a matrix?

It contains the eigenvectors

It represents the kernel of the matrix

It contains the eigenvalues

It represents the range of the matrix

38. If det(A) = 0, what does this imply about matrix A?

A is invertible

A is singular

A is symmetric

A is orthogonal

39. What is the Perron-Frobenius theorem about?

The diagonalization of matrices

The existence of eigenvalues

The properties of positive matrices

The uniqueness of the inverse

40. What is the trace of a square matrix?

Sum of the diagonal elements

Product of the diagonal elements

Determinant of the matrix

Inverse of the matrix

Matrices and Determinants MCQ Test 2