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

21. Let \(A\) and \(B\) be upper and lower triangular matrices given by

\(A=\left(\begin{array}{llll} 1& & & & * \\ & 2 & & \\ & & \ddots & \\ 0 & & & &n \end{array}\right)\) and

\(B=\left(\begin{array}{llll} 0 & & & 0 \\ & 1 & & \\ & & \ddots & \\ * & & & n-1 \end{array}\right)\)

Then

(a.) \(A\) is invertible and \(B\) is singular
(b.) \(A\) is singular and \(B\) is invertible
(c.) Both \(A\) and \(B\) are invertible
(d.) Neither \(A\) nor \(B\) is invertible

Explanation for Question 21:
NA.

22. A homogeneous system of 5 linear equations in 6 variables admits

(a.) No solution in \(\mathbf{R}^6\)
(b.) A unique solution in \(\mathbf{R}^6\)
(c.) Infinitely many solutions in \(\mathbf{R}^6\)
(d.) Finite, but more than 2 solutions in \(\mathbf{R}^6\)

Explanation for Question 22:
Note that there is a free variable.

23. Let \(A\) be an \(m \times n\) matrix with rank \(m\) and \(B\) be a \(p \times m\) matrix with rank \(p\). What will be the rank of \(B A ?(p < m < n)\)

(a.) \(m\)
(b.) \(p\)
(c.) \(n\)
(d.) \(p+m\)

Explanation for Question 23:
Let LT correspond to \(A\) and \(B\) be \(T\) and \(S\). \(T: \mathbb{R}^n \longrightarrow \mathbb{R}^m\) with \(\rho(T)=m\). \(S: \mathbb{R}^m \longrightarrow \mathbb{R}^p\) with \(\rho(S)=p\). The LT corresponding to \(BA\) is \(\Rightarrow \rho(ToS)=p\). As the composition of two onto maps is onto. This completes the explanation for Question 23.

24. Let \(A\) be an \(n \times n\) matrix and \(b=\left(b_{1}, b_{2}, \cdots, b_{n}\right)^{t}\) be a fixed vector. Consider a system of \(n\)-linear equations \(A x=b\), where \(x=\left(x_{1}, x_{2}, \cdots x_{n}\right)^{\prime}\). Consider the following statements:

A. If rank \(A=n\), the system has a unique solution

B. If rank \(A < n\), the system has infinitely many solutions

C. If \(b=0\), the system has at least one solution

Which of the following is correct?

(a.) A and B are true
(b.) A and C are true
(c.) Only A is true
(d.) Only B is true

Explanation for Question 24:
For \(A:\) If \(\rho(A)=n \Rightarrow \rho(A \mid b)=\rho(A)\) \(\Rightarrow A\) has a unique solution. For \(B:\) If \(\rho(A) < n\) and if \(\rho(A \mid b) \neq \rho(A)\) then \(A\) will have no solution. For \(C,\) \(A x=0\) has \(x=0\) as a solution. This completes the explanation for Question 24.

25. Let \(A=\left(a_{i j}\right)\) be an \(n \times n\) matrix such that \(a_{i j}=3\) for \(i\) and \(j\). Then the nullity of \(A\) is

(a.) \(n-1\)
(b.) \(n-3\)
(c.) \(n\)
(d.) 0

Explanation for Question 25:
\[ \begin{aligned} A=\left(a_{ij}\right) \text { s.t. } a_{ij}=3 \quad \\ \quad \Rightarrow \rho(A)=1 \Rightarrow \eta(A)=n-1 \end{aligned} \] This completes the explanation for Question 25.

26. Let \(A\) be a non-zero matrix of order 8 with \(A^{2}=0\). Then one of the possible values for the rank of \(A\) is

(a.) 5
(b.) 6
(c.) 4
(d.) 8

Explanation for Question 26:
If \(A^k=0\) then \(\rho(A) \leq \lfloor \frac{n(k-1)}{k} \rfloor\), where \(A\) is an \(n \times n\) square matrix.

27. The rank of the \(4 \times 4\) skew-symmetric matrix

\(\left[\begin{array}{cccc}0 & 1 & 0 & 1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ -1 & 0 & -1 & 0\end{array}\right]\) is

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

Explanation for Question 27:
Use row echelon form.

28. Let \(A\) be a \(3 \times 3\) matrix and consider the system of equations

\(A x=\left[\begin{array}{c}1 \\ 0 \\ -1\end{array}\right]\). Then

(a.) If the system is consistent then it has a unique solution
(b.) If \(A\) is singular then the system has infinitely many solutions
(c.) If the system is consistent then \(A\) is invertible
(d.) If the system has a unique solution then A is non singular

Explanation for Question 28:
If the system has a unique solution for any \(b\) then the nullspace is trivial. Hence the matrix is invertible.

29. Let \(S\) be the solution space of a set of \(m\) homogeneous linear equations with real coefficients in \(n\) unknowns. If \(A\) is the matrix of this system of equations. Then

(a.) Dimension \(S=n-\operatorname{rank} A\)
(b.) Dimension \(S\) is always \(n\)
(c.) Dimension \(S\) is infinite
(d.) Dimension \(S=n+\operatorname{rank} A\)

Explanation for Question 29:
NA

30. Let \(A\) be an \(n \times n\) matrix over \(\mathbb{R}\). Consider the following statements:

1. Rank \(A=n\)
2. Det \(A \neq 0\)

Then

(a.) 1 implies 2, but 2 does not imply 1
(b.) 2 implies 1, but 1 does not imply 2
(c.) \(1 \Leftrightarrow 2\)
(d.) There is no relation between the statements

Explanation for Question 30:
NA