Linear Algebra MCQs and MSQs with Solutions for CSIR NET, IIT JAM : System of Equations - II

11. Let \(A\) be a \(5 \times 4\) matrix with real entries such that \(A \mathbf{x}=\mathbf{0}\) if and only if \(\mathbf{x}=\mathbf{0}\) where \(\mathbf{x}\) is a \(4 \times 1\) vector and \(\mathbf{0}\) is a null vector. Then, the rank of \(A\) is

(a.) 4.
(b.) 5.
(c.) 2.
(d.) 1.

Explanation for Question 11:
\(A\) has full rank, which is 4. This completes the explanation for Question 11.

12. Let \(A=\begin{bmatrix}1 & 3 & 5 & a & 13 \\ 0 & 1 & 7 & 9 & b \\ 0 & 0 & 1 & 11 & 15\end{bmatrix}\) where \(a, b \in \mathbb{R}\).

Choose the correct statement.

(a.) There exist values of \(a\) and \(b\) for which the columns of \(A\) are linearly independent.
(b.) There exist values of \(a\) and \(b\) for which \(A \mathbf{x}=\mathbf{0}\) has \(\mathbf{x}=\mathbf{0}\) as the only solution.
(c.) For all values of \(a\) and \(b\), the rows of \(A\) span a 3-dimensional subspace of \(\mathbb{R}^{5}\).
(d.) There exist values of \(a\) and \(b\) for which \(\text{rank}(A)=2\).

Explanation for Question 12:
\(\forall a, b \in \mathbb{R}, \rho(A)=3\) \(\Rightarrow\) for \(L \cdot T\) corresponding to \(A\), \[ T: \mathbb{R}^5 \longrightarrow \mathbb{R}^3 \Rightarrow \eta(T)=2 \] Implies that \(b\) is wrong. For any values of \(a, b\), the rank of \(A\) can not be 5. Thus \(a\) and \(b\) are wrong. This completes the explanation for Question 12.

13. Let \(S\) denote the set of all primes \(p\) such that the following matrix is invertible when considered as a matrix with entries in \(\mathbb{Z} / p \mathbb{Z}\):

\(A=\begin{bmatrix}1 & 2 & 0 \\ 0 & 3 & -1 \\ -2 & 0 & 2\end{bmatrix}\)

Which of the following statements are true?

(a.) \(S\) contains all the prime numbers.
(b.) \(S\) contains all the prime numbers greater than 10.
(c.) \(S\) contains all the prime numbers other than 2 and 5.
(d.) \(S\) contains all the odd prime numbers.

Explanation for Question 13:
Note that \[ \begin{aligned} |A|=10 & \equiv 0(\bmod 2) \\ & \equiv 0(\bmod 5) \\ & \neq 0(\bmod p), p \neq 2,5 \end{aligned} \] Thus for any prime but not 2 or 5, matrix \(A\) is invertible over \(\mathbb{Z}_p\). This completes the explanation for Question 13.

14. The system of equations

\(x+y+z=1\)

\(2x+3y-z=5\)

\(x+2y-kz=4\) where \(k \in \mathbb{R}\)

has an infinite number of solutions for

(a.) \(k=0\).
(b.) \(k=1\).
(c.) \(k=2\).
(d.) \(k=3\).

Explanation for Question 14:
The matrix corresponding to the system is \[ A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 2 & 3 & -1 \\ 1 & 2 & -k \end{array}\right], b=\left[\begin{array}{l} 1 \\ 5 \\ 4 \end{array}\right] \] For the system \(A x=b\) to have infinitely many solutions \(|A|\) must be 0. If we solve \[ |A|=0 \] we get \(k=+2\). This completes the explanation for Question 14.

15. The determinant of the matrix

\(\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 & 1 & 0 \\ 2 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\)

is

(a.) 0.
(b.) -9.
(c.) -27.
(d.) 1.

Explanation for Question 15:
Do the column operation as - \(C_1 \mapsto C_1-2C_6\), \(C_1 \mapsto C_2-2C_5\), \(C_3 \mapsto C_1-2C_4\). We get an upper triangular matrix with diagonal as \(-3,-3,-3,1,1,1\).

16. Let \(A\) be a \(5 \times 4\) matrix with real entries such that the space of all solutions of the linear system

\(A X^{t}=[1,2,3,4,5]^{t}\) is given by

\(\left\{[1+2 s, 2+3 s, 3+4 s, 4+5 s]^{t}: s \in \mathbb{R}\right\}\). (Here

\(M^{t}\) denotes the transpose of a matrix \(M\)).

Then the rank of \(A\) is equal to

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

Explanation for Question 16:
Note that the system \(AX=b\) has more than one solution, thus \(A\) cannot be with full column rank, and thus the rank of \(A\) is at most 3. Also, the solution space of \(AX^t=\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{array}\right]^t\) is given by \(\{\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \end{array}\right] + \left[\begin{array}{l} 2s \\ 3s \\ 4s \\ 5s \end{array}\right] \ : s\in \mathbb{R} \}\). Which is a coset of the null space. One can easily notice that it contains two linearly independent vectors, namely \(\{(1,2,3,4),(3,5,7,9)\}\) which is obtained by putting \(s=0\) and \(s=1\). The number of LI solutions of a non-homogeneous system of equations is given by \(\eta (A)+1\), thus \(\eta (A)=1\) hence \(\rho(A)=1.\) This completes the explanation for Question 16.

17. Let \(D\) be a non-zero \(n \times n\) real matrix with \(n \geq 2\). Which of the following implications is valid?

(a.) \(\text{det}(D)=0\) implies rank\((D)=0\)
(b.) \(\text{det}(D)=1\) implies rank\((D) \neq 1\)
(c.) \(\text{det}(D)=1\) implies rank\((D) \neq 0\)
(d.) \(\text{det}(D)=n\) implies rank\((D) \neq 1\)

Explanation for Question 17:
\[ \begin{aligned} & |D|=0 \Rightarrow \operatorname{rank}(D) \neq n \\ & |D| \neq 0 \Rightarrow \operatorname{rank}(D)=n \end{aligned} \] This completes the explanation for Question 17.

18. Consider the system of \(m\) linear equations in \(n\) unknowns given by \(A x=b\), where \(A=\left(a_{i j}\right)\) is a real \(m \times n\) matrix, \(x\) and \(b\) are \(n \times 1\) column vectors. Then

(a.) There is at least one solution.
(b.) There is at least one solution if \(b\) is the zero vector.
(c.) If \(m=n\) and if the rank of \(A\) is \(n\), then there is a unique solution.
(d.) If \(m < n\) and if the rank of the augmented matrix \([A: b]\) equals the rank of \(A\), then there are infinitely many solutions.

Explanation for Question 18:
Direct results.

19. The determinant of the following \(4 \times 4\) matrix \[ A=\left[\begin{array}{cccc} 1 & 2 & 2 & 1 \\ 0 & 2 & 2 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

(a.) 4.
(b.) 6.
(c.) 12.
(d.) 24.

Explanation for Question 19:
Then \(|A|=1 \times2 \times2 \times 1=4\) This completes the explanation for Question 19.

20. The system of simultaneous linear equations \(x+y+z=0\)

\(x-y-z=0\)

Has

(a.) No solution in \(\mathbf{R}^{3}\).
(b.) A unique solution in \(\mathbf{R}^{3}\).
(c.) Infinitely many solutions in \(\mathbf{R}^{3}\).
(d.) More than 2 but finitely many solutions in \(\mathbf{R}^{3}\).

Explanation for Question 20:
The matrix corresponding to the system is \[ A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & -1 & -1 \end{array}\right] \] For a, homogeneous system always has a solution. Also, columns of \(A\) are linearly dependent, thus there are infinitely many solutions. This completes the explanation for Question 20.