Rank of Matrix MCQ

1. Let $P=\begin{bmatrix} 1 & 1 & -1\\ 2 & -3 & 4\\ 3 & -2 & 3 \end{bmatrix} and\; Q=\begin{bmatrix} -1 & -2 & -1\\ 6 & 12 & 6\\ 5 & 10 & 5 \end{bmatrix}$ be two matrices. Then the rank of P + Q is

1. 1
2. 2
3. 3
4. 0

2. The rank of the matrix $M=\begin{bmatrix} 5 & 10 & 10\\ 1 & 0 & 2\\ 3 & 6 & 6 \end{bmatrix}$ is

1. 0
2. 1
3. 2
4. 3

3. The rank of the matrix is

\[\begin{bmatrix} 1 & -1 & 0 & 0 & 0\\ 0 & 0 & 1 & -1 & 0\\ 0 & 1 & -1 & 0 & 0\\ -1 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & -1 \end{bmatrix}\]

1. 4
2. 1
3. 2
4. 3

4. Let A = [a_ij], 1 ≤ i, j ≤ n with n ≥ 3 and a_ij = i.j . The rank of A is

1. 0
2. 1
3. n – 1
4. n

5. Two matrices A and B are given below:

\[A=\begin{bmatrix} p & q\\ r & s \end{bmatrix}; B=\begin{bmatrix} p^{2}+q^{2} & pr+qs\\ pr+qs & r^{2}+s^{2} \end{bmatrix}\]

If the rank of matrix A is N, then the rank of matrix B is

1. N/2
2. N – 1
3. N
4. 2N

6. The rank of the matrix $\begin{bmatrix} 6 & 0 & 4 & 4\\ -2 & 14 & 8 & 18\\ 14 & -14 & 0 & -10 \end{bmatrix}$ is

1. 1
2. 3
3. 2
4. 0

7. If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?

1. Q will have four linearly independent rows and four linearly independent columns
2. Q will have four linearly independent rows and five linearly independent columns
3. QQ^T will be invertible
4. Q^TQ will be invertible

8. A is m x n full matrix with m > n and I is an identity matrix. Let matrix A’= (A^TA)^-1 A^T, then which one of the following statements is true?

1. AA’A= A
2. (AA’)² = A
3. AA’A = I
4. AA’A = A’

9. X = [x₁, x₂, …….., x_n]^T is an n-tuple non-zero vector. Then n x n matrix V = XX^T

1. has rank zero
2. has rank 1
3. is orthogonal
4. has rank n

10. The rank of the matrix $\begin{bmatrix} 1 & 1 & 1\\ 1 & -1 & 0\\ 1 & 1 & 1 \end{bmatrix}$ is

1. 0
2. 1
3. 2
4. 3

11. A is a 3 x 4 real matrix and Ax = b is an inconsistent system of equations. The highest possible rank of A is

1. 4
2. 1
3. 2
4. 3

12. Given matrix $\begin{bmatrix} 4 & 2 & 1 & 3\\ 6 & 3 & 4 & 7\\ 2 & 1 & 0 & 1 \end{bmatrix}$, the rank of the matrix is

1. 4
2. 2
3. 1
4. 3

13. Let A be a 4 x 3 real matrix with rank 2. Which one of the following statement is true?

1. rank of A^TA is less than 2.
2. rank of A^TA is equal to 2.
3. rank of A^TA is greater than 2.
4. rank of A^TA can be any number between 1 and 3.