Linear Algebra Test - T 2

Rsquared Mathematics

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Time: 45:00

Topic: Linear Algebra

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The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B

1. Let A, B, C, and D be square matrices of the same order then

(a.) Trace \((A B C D)=\operatorname{Trace}(B A C D)\)
(b.) Trace \((A B C D)=\operatorname{Trace}(A B D C)\)
(c.) Trace \((DBAC) $=\operatorname{Trace}(\mathrm{ABCD})\)
(d.) Trace \((C D A B)=\operatorname{Trace}(A B C D)\)

Explanation: Trace is preserved whenever order is maintained.

2. Let \(\mathrm{A}\) and \(\mathrm{B}\) be two square matrices of order \(n\) such that \(A B+A^2=I-A B^2\), then

(a.) \(B\) is non-singular
(b.) \(A\) is non-singular
(c.) Both \(A\) and \(B\) are non-singular
(d.) Neither \(A\) nor \(B\) need to be non-singular.

Explanation: \(A B+A^2=I-A B^2, AB(I + B) = I\). Hence \(AB\) must be non-singular.

3. Which of the following is a subspace of the vector space of all \(n \times n\) matrices over a field of real numbers -

(a.) \(W=\left\{A=\left[a_{i j}\right]_{n \times n} ; a_{11}^2+a_{22}^2=1-a_{33}^2\right\} \)
(b.) \(W=\left\{A=\left[a_{i j}\right]_{n \times n} ; a_{11}=a^3, a_{22}=b^3, a, b \in \mathbb{R}\right\}\)
(c.) \(W=\{A=\left[a_{i j}\right]_{n \times n} ; A\) is Idempotent\(\}\)
(d.) \(W=\{A=\left[a_{i j}\right]_{n \times n} ; A\) is nilpotent matrix of index \(\leq 2\}\)

Explanation: Every real number can be written as a cube of some real number uniquely.

4. Let \(\mathrm{A}\) be a real orthogonal matrix of order \(2 \times 2\) such that \(\operatorname{Trace}(\mathrm{A})=1\), given that \(\mathrm{u}=(1,2) \in \mathbb{R}^2\) and \(\mathrm{v}=A u^T\), then the angle between \(\mathrm{u}\) and \(\mathrm{v}\) is -

(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) Can not be found from the given information

Explanation: \(A\) is a rotation matrix with an angle of 60 degrees.

5. Let \(\mathrm{M}=\left(\begin{array}{ccc}0 & 1 & -2 \\ -1 & 0 & \alpha \\ 2 & -\alpha & 0\end{array}\right), \alpha \in \mathbb{R}-\{0\}\) and let \(\mathrm{b}\) be a nonzero vector such that \(\mathrm{Mx}=\mathrm{b}\) for some \(\mathrm{x}\). Then the value \(x^T b\) is

(a.) \(-\alpha\)
(b.) \(\alpha\)
(c.) 0
(d.) 1

Explanation: \(M\) is skew-symmetric.

Part - C

6. Let \(W\) be a subspace over \(\mathbb{R}\) and let \(T: \mathbb{R}^6 \rightarrow W\) be a linear transformation such that \(S=\left\{T e_1, T e_3, T e_5\right\}\) spans \(W\)

(a.) \(S\) is a basis of \(W\)
(b.) \(\text{Ker}(T)\) contains more than one element
(c.) \(T(\mathbb{R}^6) \neq W\)
(d.) \(S=\left\{T e_2, T e_4, T e_6\right\}\) spans \(W\)

Explanation: Dimension \(\text{dim}(W) < 6\).

7. Let \(V\) be a vector space of dimension \(n\) over a finite field with \(k\) elements. Then the cardinality of \(V\) is -

(a.) Infinite
(b.) \(k^n\)
(c.) \(n^k\)
(d.) \(\frac{n!}{k!(n-k)!}\)

Explanation: \(|V| = |F|^{dim(V)}\)

8. \(V=\left\{A=\left[a_{i j}\right]_{n \times n} ; A B=B A, \forall B\right\}\) then the dimension of \(V\) over the field \(\mathbb{R}\)

(a.) \(n\)
(b.) \(\sum n\)
(c) 1
(d.) \(V\) does not form a vector space over \(\mathbb{R}\)

Explanation: Only the scalar matrix commutes with all matrices.

9. Let \(C(a, b)\) be the vector space of all continuous functions defined on an open interval \((a, b)\) over the field \(\mathbb{R}\) then which of the following is/are not a subspace of \(C(a, b)\) -

(a.) \(\left\{f \in C(a, b): \lim _{x \rightarrow \frac{a+b}{2}} f(x)=0\right\}\)
(b.) \(\left\{f \in C(a, b): \lim _{x \rightarrow \frac{a+b}{2}} f(x)=1\right\}\)
(c.) \(\{f \in C(-\infty, \infty): f(x)=f(-x)\}\)
(d.) All of the above

Explanation: Set of all even functions forms a vector space.

10. Which of the following are correct?

(a.) \(A\) and \(B\) are similar then \(C_A(x)=C_B(x)\)
(b.) If \(C_A(x)=C_B(x)\) then \(A\) and \(B\) are similar
(c.) If \(A=B^2\) then \(C_A(x)=(C_B(x))^2\)
(d.) If \(A=B^2\) then \(C_A(x)-C_{B^2}(x)=C_0(x)\)

Explanation: Similar matrices have the same characteristic polynomial.