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

$A=\left(\begin{array}{lllll}1& & & & \ast \\ & 2& & \\ & & \ddots & \\ 0& & & & n\end{array}\right)$ and

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

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 $BA?(p<m<n)$

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

Explanation for Question 23:
Let LT correspond to $A$ and $B$ be $T$ and $S$. $T:{\mathbb{R}}^n⟶{\mathbb{R}}^m$ with $\rho (T)=m$. $S:{\mathbb{R}}^m⟶{\mathbb{R}}^p$ 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 $b={(b_1,b_2,\cdots ,b_n)}^t$ be a fixed vector. Consider a system of $n$-linear equations $Ax=b$, where $x={(x_1,x_2,\cdots x_n)}^{\prime }$. 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 $\rho (A\mid b)\ne \rho (A)$ then $A$ will have no solution. For $C,$ $Ax=0$ has $x=0$ as a solution. This completes the explanation for Question 24.

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

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

Explanation for Question 25:
$$\begin{array}{r}A=\left(a_{ij}\right)\text{ s.t. }{a}_{ij}=3\\ \Rightarrow \rho (A)=1\Rightarrow \eta (A)=n-1\end{array}$$ 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 $\rho (A)\le \lfloor \frac{n(k-1)}{k}\rfloor$, 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

$Ax=\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]$. 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 $S=n-\mathrm{rank}A$
(b.) Dimension $S$ is always $n$
(c.) Dimension $S$ is infinite
(d.) Dimension $S=n+\mathrm{rank}A$

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 $A\ne 0$

Then

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

Explanation for Question 30:
NA