1 $If matrix is of order and matrix is of order, then the order of matrix is:$

A.

B.

C.

D.

2 $Which of the following is true for the transpose of the product of two matrices ?$

A.

B.

C.

D.

3 $A square matrix is said to be symmetric if:$

A.

B.

C.

D.

4 $The trace of a matrix is defined as:$

A.

The sum of the principal diagonal elements

B.

The sum of all elements

C.

The determinant of the matrix

D.

The product of the diagonal elements

5 $Which elementary row operation changes the value of the determinant of a matrix?$

A.

Multiplying a row by 1

B.

Interchanging two rows ()

C.

None of the above

D.

Adding a multiple of one row to another ()

6 $The rank of a matrix is defined as the order of the largest square sub-matrix whose determinant is:$

A.

Not zero

B.

Negative

C.

Zero

D.

Positive

7 $What is the rank of the identity matrix of order ?$

A.

B.

$0$

C.

$1$

D.

8 $Find the rank of the matrix$

A.

0

B.

4

C.

2

D.

1

9 $If a matrix of order is singular (), then its rank is:$

A.

Exactly 3

B.

Less than 3

C.

Greater than 3

D.

Exactly 0

10 $The rank of a matrix is equal to:$

A.

The number of non-zero rows in its Echelon form

B.

The number of rows

C.

The number of columns

D.

The sum of diagonal elements

11 $A set of vectors is linearly dependent if there exist scalars (not all zero) such that:$

A.

B.

C.

D.

12 $Two vectors in, and, are:$

A.

Orthonormal

B.

Orthogonal

C.

Linearly Independent

D.

Linearly Dependent

13 $If the determinant of a matrix formed by vectors as columns is non-zero, then the vectors are:$

A.

Linearly Dependent

B.

Zero vectors

C.

Linearly Independent

D.

Parallel

14 $What is the maximum number of linearly independent vectors in ?$

A.

B.

C.

D.

15 $A system of linear equations is consistent if and only if:$

A.

B.

C.

D.

16 $For a system of linear equations in variables, if, then the system has:$

A.

A unique solution

B.

Only the trivial solution

C.

No solution

D.

Infinitely many solutions

17 $For a system of linear equations in variables, if, then the system has:$

A.

A unique solution

B.

Infinitely many solutions

C.

Exactly two solutions

D.

No solution

18 $The system of equations has no solution if:$

A.

B is a zero vector

B.

C.

D.

19 $A homogeneous system of equations always has:$

A.

Infinite solutions

B.

No solution

C.

Only non-trivial solutions

D.

At least one solution (Trivial solution)

20 $A homogeneous system in unknowns has a non-trivial solution if:$

A.

B.

C.

D.

A is an identity matrix

21 $Consider the system: and . This system is:$

A.

Consistent with infinite solutions

B.

Inconsistent

C.

Homogeneous

D.

Consistent with unique solution

22 $For what value of does the system,, have only the trivial solution?$

A.

B.

C.

Any real number

D.

23 $A square matrix is invertible (non-singular) if and only if:$

A.

B.

C.

D.

24 $The inverse of a matrix is given by:$

A.

B.

C.

D.

25 $If and are invertible matrices, then is equal to:$

A.

B.

C.

D.

26 $The inverse of an orthogonal matrix is:$

A.

B.

C.

D.

27 $If, then is:$

A.

B.

C.

D.

28 $The roots of the characteristic equation are called:$

A.

Latent vectors

B.

Eigenvalues

C.

Rank

D.

Eigenvectors

29 $If is an eigenvalue of and is the corresponding eigenvector, then:$

A.

B.

C.

D.

30 $Find the eigenvalues of the matrix$

A.

2, 3

B.

1, 3

C.

0, 3

D.

1, 2

31 $The sum of the eigenvalues of a matrix is equal to:$

A.

Determinant of the matrix

B.

Trace of the matrix

C.

Highest eigenvalue

D.

Zero

32 $The product of the eigenvalues of a matrix is equal to:$

A.

Trace of the matrix

B.

Rank of the matrix

C.

1

D.

Determinant of the matrix

33 $If the eigenvalues of a matrix are 2, 3, and 4, then the eigenvalues of are:$

A.

B.

2, 3, 4

C.

4, 9, 16

D.

1/2, 1/3, 1/4

34 $If is an eigenvalue of a non-singular matrix, then the eigenvalue of is:$

A.

B.

C.

D.

35 $The eigenvalues of a real symmetric matrix are always:$

A.

0

B.

Real

C.

Complex with non-zero imaginary part

D.

Purely imaginary

36 $The eigenvalues of a skew-symmetric matrix are:$

A.

Either 0 or purely imaginary

B.

Always 1

C.

Always real

D.

Always positive

37 $At least one eigenvalue of a singular matrix is:$

A.

Infinite

B.

-1

C.

1

D.

0

38 $If has eigenvalues 2 and 5, what are the eigenvalues of ?$

A.

2, 5

B.

5, 8

C.

-1, 2

D.

6, 15

39 $What are the eigenvalues of the matrix ?$

A.

0, 0

B.

-1, -1

C.

1, -1

D.

1, 1

40 $The Cayley-Hamilton theorem states that every square matrix satisfies its own:$

A.

Inverse

B.

Diagonal elements

C.

Transpose

D.

Characteristic equation

41 $If the characteristic equation of a matrix is, then according to Cayley-Hamilton theorem:$

A.

B.

C.

D.

42 $The Cayley-Hamilton theorem can be used to find:$

A.

Trace only

B.

Determinant only

C.

Inverse of a matrix

D.

Rank of a matrix

43 $Given, the inverse is given by:$

A.

B.

C.

D.

44 $An eigenvector corresponding to an eigenvalue must be:$

A.

A non-zero vector

B.

A zero vector

C.

A unit vector

D.

A column of the identity matrix

45 $Algebraic Multiplicity of an eigenvalue refers to:$

A.

The rank of the matrix

B.

The value of the eigenvalue itself

C.

The number of times the eigenvalue appears as a root of the characteristic equation

D.

The number of linearly independent eigenvectors associated with it

46 $A square matrix of order is diagonalizable if and only if:$

A.

It has distinct eigenvalues

B.

It has linearly independent eigenvectors

C.

Its determinant is non-zero

D.

It is symmetric

47 $If, its eigenvalues are:$

A.

0, 0

B.

-1, -1

C.

1, 0

D.

1, 1

48 $Which of the following matrices is in Row Echelon Form?$

A.

B.

C.

D.

49 $If is a matrix with eigenvalues 1, -1, and 2, the determinant of is:$

A.

0

B.

-2

C.

2

D.

3

50 $If is a matrix with eigenvalues 1, -1, and 2, the Trace of is:$

A.

2

B.

0

C.

-2

D.

3

Unit 1 - Practice Quiz