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 \(A x=0\)?

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

Explanation for Question 1:
Let \(T\) be the linear transformation corresponding to \(A\), say \(T: \mathbb{F}^{50} \rightarrow \mathbb{F}^{20}\) 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.) \(A x=b\) has a solution for any \(b\).
(b.) \(A x=0\) does not have a solution.
(c.) If \(A x=b\) 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:
\[ \begin{aligned} A=\left[\begin{array}{lll} 4 & 8 & 4 \\ 3 & 6 & 1 \\ 2 & 4 & 0 \end{array}\right] & \sim\left[\begin{array}{lll} 4 & 8 & 4 \\ 3 & 6 & 1 \\ 1 & 2 & 0 \end{array}\right] \sim\left[\begin{array}{lll} 0 & 0 & 4 \\ 0 & 0 & 1 \\ 1 & 2 & 0 \end{array}\right] \\ & \sim\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 2 & 0 \end{array}\right] \end{aligned} \] This completes the explanation for Question 3.

4. The determinant of the \(n \times n\) permutation matrix

\(\begin{pmatrix} & & & & & 1 \\ & & & & 1 & \\ & & & \cdot & & \\ & & \cdot & & & \\ & 1 & & & & \\ 1 & & & & & \end{pmatrix}\)

is

(a.) \((-1)^n\)
(b.) \((-1)^{\left\lceil\frac{n}{2}\right\rceil}\)
(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.) \(\left(x^{2}-y^{2}\right)\left(y^{2}-z^{2}\right)\left(z^{2}-x^{2}\right)\)

Explanation for Question 5:
\[ \begin{aligned} & =z^2 y-y^2 z+x y^2-x y z-x z^2+x^2 z-y x^2+x y z \\ & =y\left(z^2-y^0 z+x y-x_2\right)-x\left(z^2+y x-x^2 z-y^2\right) \\ & =\Rightarrow(y-x)\left(z^2-y z+x y-x z\right) \\ & =(y-x)(z-x) \lg 2)(z-y) \\ & \end{aligned} \] 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 \(A x=b\) has a complex solution then

(a.) \(A x=b\) has an integer solution.
(b.) \(A x=b\) has a rational solution.
(c.) The set of real solutions to \(A x=0\) has a basis consisting of rational solutions.
(d.) If \(b \neq 0\) 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 \(A=\begin{pmatrix} 5 & 9 & 8 \\ 1 & 8 & 2 \\ 9 & 1 & 0 \end{pmatrix}\) 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 \(A B=I_{4}\) 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.) \(B A=I_{7}\) 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 \(T_A: \mathbb{R}^{7} \rightarrow \mathbb{R}^{4}\). 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 \(T_B: \mathbb{R}^{4} \rightarrow \mathbb{R}^{7}.\) 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.) \(A=\begin{pmatrix} \cos\frac{\pi}{4} & \sin\frac{\pi}{4} & 0 \\ -\sin\frac{\pi}{4} & \cos\frac{\pi}{4} & 0 \\ 0 & 0 & 1 \end{pmatrix}\)
(b.) \(A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\frac{\pi}{3} & \sin\frac{\pi}{3} \\ 0 & -\sin\frac{\pi}{3} & \cos\frac{\pi}{3} \end{pmatrix}\)
(c.) \(A=\begin{pmatrix} \cos\frac{\pi}{6} & 0 & \sin\frac{\pi}{6} \\ 0 & 1 & 0 \\ -\sin\frac{\pi}{6} & 0 & \cos\frac{\pi}{6} \end{pmatrix}\)
(d.) \(A=\begin{pmatrix} \cos\frac{\pi}{2} & \sin\frac{\pi}{2} & 0 \\ -\sin\frac{\pi}{2} & \cos\frac{\pi}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}\)

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 \(A x=0\) where \(A\) is an \(m \times n\) real matrix and \(n>m\). Which of the following statements are always true?

(a.) \(A x=0\) has a solution.
(b.) \(A x=0\) has no nonzero solution.
(c.) \(A x=0\) 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 \(T: \mathbb{R}^n \longrightarrow \mathbb{R}^m\) s.t. \(m < n\) . Using the rank-nullity theorem, \[ \begin{gathered} \eta(T)>0 \\ \Rightarrow \operatorname{Ker}(T) \neq\{0\} \end{gathered} \] \(\Rightarrow A x=0\) has a nonzero solution and the nullity of \(T\) is at least \(n-m\). This completes the explanation for Question 10.