Matrix Algebra MCQs

1. What is the rank of the matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \) ?

a) 1 b) 2 c) 3 d) 0

Correct Answer: b) 2

Explanation: The rank of a matrix is the number of non-zero rows in its row echelon form. Here, the third row is a linear combination of the first two, giving a rank of 2.

2. Find the inverse of \( \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \) .

a) \( \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} \) b) \( \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} \) c) \( \begin{bmatrix} 2 & -3 \\ -1 & 4 \end{bmatrix} \) d) \( \begin{bmatrix} 4 & -1 \\ -3 & 2 \end{bmatrix} \)

Correct Answer: a) \( \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} \)

Explanation: The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by: \[ \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] Applying this formula gives the correct answer.

3. The system \(x + y = 5\), \(2x + 2y = 10\) has:

a) No solution b) Exactly one solution c) Infinitely many solutions d) Exactly two solutions

Correct Answer: c) Infinitely many solutions

Explanation: The second equation is simply a multiple of the first equation. Both equations represent the same line. Therefore, every point on the line \(x + y = 5\) is a solution, resulting in infinitely many solutions.

4. If a 4×5 matrix has rank 3, how many free variables are there in its solution?

a) 1 b) 2 c) 3 d) 4

Correct Answer: b) 2

Explanation: The number of free variables is given by the formula: \[ \text{Free Variables} = \text{Number of Columns} - \text{Rank} \] Here, there are 5 columns and the rank is 3. \[ \text{Free Variables} = 5 - 3 = 2 \]

5. The inverse of \( \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \) is:

a) \( \begin{bmatrix} 0.5 & 0 \\ 0 & 0.33 \end{bmatrix} \) b) \( \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \) c) \( \begin{bmatrix} 0 & 2 \\ 3 & 0 \end{bmatrix} \) d) \( \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix} \)

Correct Answer: a) \( \begin{bmatrix} 0.5 & 0 \\ 0 & 0.33 \end{bmatrix} \)

Explanation: The inverse of a diagonal matrix is simply the reciprocal of its diagonal elements. Therefore, the inverse is: \( \begin{bmatrix} 1/2 & 0 \\ 0 & 1/3 \end{bmatrix} \).

6. For what value of \( k \) does the system \( x + ky = 2 \), \( 3x + 6y = 6 \) have no solution?

a) 1 b) 2 c) 3 d) 4

Correct Answer: a) 1

Explanation: The second equation is equivalent to \( x + 2y = 2 \). For no solution, the system must be inconsistent. If \( k = 1 \), the first equation becomes \( x + y = 2 \) and the second remains \( x + 2y = 2 \). These lines are parallel and distinct, resulting in no solution.

7. What is the rank of a 3×3 identity matrix?

a) 0 b) 1 c) 2 d) 3

Correct Answer: d) 3

Explanation: The identity matrix has full rank equal to its dimension (3), as all rows/columns are linearly independent.

8. If A is invertible, which statement is FALSE?

a) A has full rank b) det(A) ≠ 0 c) Ax = 0 has infinitely many solutions d) The RREF of A is I

Correct Answer: c) Ax = 0 has infinitely many solutions

Explanation: For invertible A, Ax=0 has only the trivial solution x=0. The other options are true properties of invertible matrices.

9. The solution to the system 2x + y = 5, x - y = 1 is:

a) (1, 2) b) (2, 1) c) (1, 3) d) (3, 1)

Correct Answer: b) (2, 1)

Explanation: Adding the equations gives 3x=6 ⇒ x=2. Substituting into second equation: 2-y=1 ⇒ y=1.

10. If A is a 5×7 matrix with rank 4, what is the dimension of the null space?

a) 1 b) 2 c) 3 d) 4

Correct Answer: c) 3

Explanation: By the Rank-Nullity Theorem: nullity = number of columns (7) - rank (4) = 3.

Matrix Algebra MCQs

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