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

Answer
Answer. b

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

Answer
Answer. c

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

Answer
Answer. a

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

  1. 0
  2. 1
  3. n – 1
  4. n

Answer
Answer. b

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

Answer
Answer. c

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

Answer
Answer. c

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. QQT will be invertible
  4. QTQ will be invertible

Answer
Answer. a

8. A is m x n full matrix with m > n and I is an identity matrix. Let matrix A’= (ATA)-1 AT, 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’

Answer
Answer. a

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

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

Answer
Answer. a

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

Answer
Answer. c

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

Answer
Answer. c

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

Answer
Answer. b

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

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

Answer
Answer. b

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