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

1. The row space of a $20\times 50$ matrix $A$ has dimension 13. What is the dimension of the space of solutions of $Ax=0$?

(a.) 7
(b.) 13
(c.) 33
(d.) 37

Explanation for Question 1:
Let $T$ be the linear transformation corresponding to $A$, say $T:{\mathbb{F}}^{50}\to {\mathbb{F}}^{20}$ has rank 13. Then by the rank nullity theorem, $$n(T)=37$$ This completes the explanation for Question 1.

2. Let $A$ be an $m\times n$ matrix of rank $n$ with real entries. Choose the correct statement.

(a.) $Ax=b$ has a solution for any $b$.
(b.) $Ax=0$ does not have a solution.
(c.) If $Ax=b$ has a solution, then it is unique.
(d.) $y^{\prime }A=0$ for some nonzero $y$, where $y^{\prime }$ denotes the transpose of the vector $y$.

Explanation for Question 2:
Note that the corresponding linear transformation to $A$ is one-one as it has a full column rank. Thus, if a solution exists, then it is unique. This completes the explanation for Question 2.

3. Which of the following matrices has the same row space as the matrix $\left(\begin{array}{lll}4& 8& 4\\ 3& 6& 1\\ 2& 4& 0\end{array}\right)$?

(a.) $\left(\begin{array}{lll}1& 2& 0\\ 0& 0& 1\end{array}\right)$
(b.) $\left(\begin{array}{lll}1& 1& 0\\ 0& 0& 1\end{array}\right)$
(c.) $\left(\begin{array}{lll}0& 1& 0\\ 0& 0& 1\end{array}\right)$
(d.) $\left(\begin{array}{lll}1& 0& 0\\ 0& 1& 0\end{array}\right)$

Explanation for Question 3:
$$\begin{array}{rl}A=\left[\begin{array}{lll}4& 8& 4\\ 3& 6& 1\\ 2& 4& 0\end{array}\right]& \sim \left[\begin{array}{lll}4& 8& 4\\ 3& 6& 1\\ 1& 2& 0\end{array}\right]\sim \left[\begin{array}{lll}0& 0& 4\\ 0& 0& 1\\ 1& 2& 0\end{array}\right]\\ & \sim \left[\begin{array}{lll}0& 0& 1\\ 0& 0& 1\\ 1& 2& 0\end{array}\right]\end{array}$$ This completes the explanation for Question 3.

4. The determinant of the $n\times n$ permutation matrix

$\left(\begin{array}{cccccc}& & & & & 1\\ & & & & 1& \\ & & & \cdot & & \\ & & \cdot & & & \\ & 1& & & & \\ 1& & & & & \end{array}\right)$

is

(a.) $(-1{)}^n$
(b.) $(-1{)}^{\lceil \frac{n}{2}\rceil }$
(c.) -1
(d.) 1

Explanation for Question 4:
For $n=1,|A|=1\Rightarrow a,c$ are false. For $n=2,|A|=-1\Rightarrow d$ is false. This completes the explanation for Question 4.

5. The determinant

$\left|\begin{array}{lll}1& 1+x& 1+x+x^2\\ 1& 1+y& 1+y+y^2\\ 1& 1+z& 1+z+z^2\end{array}\right|$

is equal to

(a.) $(z-y)(z-x)(y-x)$
(b.) $(x-y)(x-z)(y-z)$
(c.) $(x-y{)}^2(y-z{)}^2(z-x{)}^2$
(d.) $(x^2-y^2)(y^2-z^2)(z^2-x^2)$

Explanation for Question 5:
$$\begin{array}{rl}& =z^{2}y-y^{2}z+x{y}^2-xyz-x{z}^2+x^{2}z-y{x}^2+xyz\\ & =y(z^2-y^{0}z+xy-x_2)-x(z^2+yx-x^{2}z-y^2)\\ & =\Rightarrow (y-x)(z^2-yz+xy-xz)\\ & =(y-x)(z-x)\lg 2)(z-y)\\ & \end{array}$$ This completes the explanation for Question 5.

6. Let $A$ be a $3\times 4$ matrix and $b$ a $3\times 1$ matrix with integer entries. Suppose that the system $Ax=b$ has a complex solution then

(a.) $Ax=b$ has an integer solution.
(b.) $Ax=b$ has a rational solution.
(c.) The set of real solutions to $Ax=0$ has a basis consisting of rational solutions.
(d.) If $b\ne 0$ then $A$ has positive rank.

Explanation for Question 6:
Note that the system is consistent and entries in $A$ are integers implying that the solution is rational but need not be an integer. Also, if $b$ is non-zero then matrix $A$ must be non-zero, hence the options b, c, and d are correct. This completes the explanation for Question 6.

7. The matrix $A=\left(\begin{array}{ccc}5& 9& 8\\ 1& 8& 2\\ 9& 1& 0\end{array}\right)$ satisfies:

(a.) $A$ is invertible and the inverse has all integer entries.
(b.) Det$(A)$ is odd.
(c.) Det$(A)$ is divisible by 13.
(d.) Det$(A)$ has at least two prime divisors.

Explanation for Question 7:
$|A|=-416=-13\times 32$ $A^{-1}$ does not have integer entries as there are entries in $adj(A)$ which are not divisible by 13. This completes the explanation for Question 7.

8. Let $A$ be a $4\times 7$ real matrix and $B$ be a $7\times 4$ real matrix such that $AB=I_4$ where $I_4$ is the $4\times 4$ identity matrix. Which of the following is/are always true?

(a.) Rank$(A)=4$.
(b.) Rank$(B^{\prime })=7$.
(c.) Nullity$(B)=0$.
(d.) $BA=I_7$ where $I_7$ is the $7\times 7$ identity matrix.

Explanation for Question 8:
As the rank of $AB$ is 4, it implies that the rank of $A$ and $B$ is at least 4. Let the linear transformations corresponding to $A$ be $T_A:{\mathbb{R}}^7\to {\mathbb{R}}^4$. Thus, using the rank-nullity theorem, the rank of $A$ can be at most 4. Combining together, the rank of $A$ must be 4. Let the linear transformations corresponding to $B$ be $T_B:{\mathbb{R}}^4\to {\mathbb{R}}^7.$ Note that the rank of $B$ is almost 4. Thus, the rank of $B$ is also 4 and $T_B$ is a one-one linear transformation. Hence the nullity of $B$ is $0$. This completes the explanation for Question 8.

9. For the matrix $A$ as given below, which of them satisfy $A^6=I$?

(a.) $A=\left(\begin{array}{ccc}\cos \frac{\pi }{4}& \sin \frac{\pi }{4}& 0\\ -\sin \frac{\pi }{4}& \cos \frac{\pi }{4}& 0\\ 0& 0& 1\end{array}\right)$
(b.) $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \frac{\pi }{3}& \sin \frac{\pi }{3}\\ 0& -\sin \frac{\pi }{3}& \cos \frac{\pi }{3}\end{array}\right)$
(c.) $A=\left(\begin{array}{ccc}\cos \frac{\pi }{6}& 0& \sin \frac{\pi }{6}\\ 0& 1& 0\\ -\sin \frac{\pi }{6}& 0& \cos \frac{\pi }{6}\end{array}\right)$
(d.) $A=\left(\begin{array}{ccc}\cos \frac{\pi }{2}& \sin \frac{\pi }{2}& 0\\ -\sin \frac{\pi }{2}& \cos \frac{\pi }{2}& 0\\ 0& 0& 1\end{array}\right)$

Explanation for Question 9:
All matrices are rotation matrices $A^6=I\Rightarrow$ Angle of rotation can be $\frac{2\pi }{6},\frac{2\pi }{3},\frac{2\pi }{2}$ $\Rightarrow a,c,d$ are false. This completes the explanation for Question 9.

10. Consider a homogeneous system of linear equations $Ax=0$ where $A$ is an $m\times n$ real matrix and $n>m$. Which of the following statements are always true?

(a.) $Ax=0$ has a solution.
(b.) $Ax=0$ has no nonzero solution.
(c.) $Ax=0$ has a nonzero solution.
(d.) Dimension of the space of all solutions is at least $n-m$.

Explanation for Question 10:
Linear transformation corresponding to $A$, say $T:{\mathbb{R}}^n⟶{\mathbb{R}}^m$ s.t. $m<n$ . Using the rank-nullity theorem, $$\begin{array}{c}\eta (T)>0\\ \Rightarrow \mathrm{Ker}(T)\ne \{0\}\end{array}$$ $\Rightarrow Ax=0$ has a nonzero solution and the nullity of $T$ is at least $n-m$. This completes the explanation for Question 10.