50 questions on linear algebra for NET and GATE aspirants - Gonit Sora (গণিত চ'ৰা)

22 Jun 50 questions on linear algebra for NET and GATE aspirants

Posted at 19:32h in Articles, Careers, English, NET / GATE / SET, Problems by Pankaj Jyoti Mahanta

Find the correct options:

1) $M=\left(\begin{array}{ccc}1 & 2 & 2 \\0 & 2 & 2 \\0 & 1 & 1 \end{array}\right)$ and $V={ Mx^{T} : x\in R^{3}}.$ Then $dim V$ is

(a) 0 (b) 1 (c) 2 (d) 3

2) $A^{2}-A=0,$ where $A$ is a $9\times 9$ matrix. Then

(a) A must be a zero matrix (b) A is an identity matrix

3) The number of linearly independent eigen vectors of $\left(\begin{array}{cccc}1 & 1 & 0 & 0 \\2 & 2 & 0 & 0 \\0 & 0 & 3 & 0\\0 & 0 & 5 & 5 \end{array}\right)$ is

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

4) The minimal polynomial of $\left(\begin{array}{cccc}2 & 1 & 0 & 0 \\0 & 2 & 0 & 0 \\0 & 0 & 2 & 0\\0 & 0 & 0 & 5 \end{array}\right)$ is

(a) $(x-2)$ (b) $(x-2)(x-5)$ (c) $(x-2)^{2}(x-5)$ (d) $(x-2)^{3}(x-5)$

5) A is a unitary matrix. Then eigen value of A are

(a) 1, -1 (b) 1, -i (c) i, -i (d) -1, i

6) $\left(\begin{array}{ccc}2 & -3 \\2 & -2 \end{array}\right)$ is an operator on $R^{2}.$ The invariant subspaces of the operator are

(a) $R^{2}$ and the subspace with base {(0,1)} (b) $R^2$ and the zero subspace

7) Rank of the matrix $\left(\begin{array}{ccccc}21 & -7 & 0 & 0 & 0 \\-11 & 9 & 0 & 0 & 0 \\0 & -19 & 35 & 0 & 0 \\0 & 15 & 0 & 12 & 20 \\0 & 0 & -24 & 21 & 35 \end{array}\right)$ is

(a) 2 (b) 3 (c) 4 (d) 5

8) The dimension of the subspace of $M_{2\times 2}$ spanned by $\left(\begin{array}{ccc}1 & -5 \\-4 & 2 \end{array}\right),$ $\left(\begin{array}{ccc}1 & 1 \\-1 & 5 \end{array}\right)$ and $\left(\begin{array}{ccc}2 & -4 \\-5 & 7 \end{array}\right)$ is

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

9) U and V are subspace of $R^{4}$ such that

U = span [(1,2,3,4), (5,7,2,1), (3,1,4,-3)]

V=span [(2,1,2,3), (3,0,1,2), (1,1,5,3)].

Then the dimension of $U\cap V$ is

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

10) Let $M_{n\times n}$ be the set of all n-square symmetric matrices and the characteristics polynomial of each $A\in M_{n\times n}$ is of the form

$t^{n}+t^{n-2}+a_{n-3}t^{n-3}+\dots +a_{1}t+a_{0}.$ Then the dimension of $M_{n\times n}$ over R is

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

11) A is a $3\times 3$ matrix with $\sigma (A)={1, -1, 0 }.$ Then $|I+A^{100}|$ is

(a) 6 (b) 4 (c) 9 (d) 100

12) A is $5\times 5$ matrix, all of whose entries are 1, then

(a) A is not diagonalizable (b) A is idempotent (c) A is nilpotent

(d) The minimal polynomial and the characteristics polynomial of A are not equal.

13) A is an upper triangular with all diagonal entries zero, then I+A is

(a) invertible (b) idempotent (c) singular (d) nilpotent

14) Number of linearly independent eigen vectors of $\left(\begin{array}{cccc}2 & 2 & 0 & 0 \\2 & 1 & 0 & 0 \\0 & 0 & 3 & 0 \\0 & 0 & 1 & 4 \end{array}\right)$ is

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

15) A is a $5\times 5$ matrix over $R,$ then $(t^{2}+1)(t^{2}+2)$

(a) is a minimal polynomial (b) is a characteristics polynomial

16) M is a 2-square matrix of rank 1, then M is

(a) diagonalizable and non singular (b) diagonalizable and nilpotent

17) A be a n-square matrix with integer entries and $B=A+\frac{1}{2} I.$ Then

(a) B is idempotent (b) $B^{-1}$ exist (c) B is nilpotent (d) B-I is idempotent

18) Let $A\in M_{3\times 3}(R),$ then $t^{2}+1$ is

(a) a minimal polynomial of A (b) a characteristics polynomial of A

19) A is a 4-square matrix and $A^{5}=0.$ Then

(a) $A^{4}=I$ (b) $A^{4}=A$ (c) $A^{4}=0$ (d) $A^{4}=-I$

20) $A=\left(\begin{array}{ccc}0 & 1 & a \\0 & 0 & 1 \\0 & 0 & 0 \end{array}\right)$ and $B=\left(\begin{array}{ccc}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0 \end{array}\right).$ Then

(a) A and B are similar (b) A and B are not similar

21) Let $S={ 2-x+3x^{2}, x+x^{2}, 1-2x^{2} }$ be subset of $P_{2}(R).$ Then

(a) S is linearly independent (b) S is linearly dependent

22) $T: P_{2}(R)\rightarrow P_{3}(R)$ such that $T(f(x))=2f^{ '}(x)+3 \int_{0}^{x}f(t)dt.$ Then rank of T is

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

23) $T_{i}: P(R)\rightarrow P(R)$ such that $T_{1}(f(x))= \int_{0}^{x}f(t)dt$ and $T_{2}(f(x))=f^{'}(x).$ Then

(a) $T_{1}$ is 1-1, $T_{2}$ is not (b) $T_{2}$ is 1-1, $T_{1}$ is not

24) $T: P_{3}(R)\rightarrow M_{2\times 2}(R),$ such that $T(f(x))= \left(\begin{array}{ccc}f(1) & f(2)\\f(3) & f(4)\end{array}\right),$ then

(a) T is 1-1 (b) T is onto (c) T is both 1-1 and onto (d) T is neither 1-1 nor onto

25) W=L{(1,0,0,0), (0,1,0,0)}, then

(a) $\frac{R^{4}}{W}=L{ W+(2,0,0,0), W+(0,2,0,0) }$

(b) $\frac{R^{4}}{W}=L{ W+(1,2,3,4), W+(2,3,4,5) }$

(d) ${ W+(1,2,3,4), W+(2,3,4,5) }$ is a basis of $\frac{R^{4}}{W}$

26) $A= \left(\begin{array}{ccc}0 & 1 \\0 & 0 \end{array}\right),$ then

(a) A has zero image (b) all the eigen value of A are zero

27) $T:R^{4}\rightarrow R^{4},$ defined by $T(e_{1})=e_{2}, T(e_{2})=e_{3}, T(e_{3})=0, T(e_{4})=e_{3}.$ Then

(a) T is nilpotent (b) T has at least one non-zero eigen value

28) $A= \left(\begin{array}{ccc}a & a & a \\a & a & a \\a & a & a \end{array}\right),$ where $a\neq 0,$ then

(a) A is not diagonalizable (b) A is idempotent

29) $A\in M_{2\times 2}(R)$ and rank of A is 1, then

(a) A is diagonalizable (b) A is nilpotent

30) A is a 3-square matrix and the eigen values of A are -1, 0, 1 with respect to the eigen vectors $(1,-1,0)^{T}, (1,1,-2)^{T}, (1,1,1)^{T}.$ then 6A is

(a) $\left(\begin{array}{ccc}1 & 5 & 3 \\5 & 1 & 3 \\3 & 3 & 3 \end{array}\right)$

(b) $\left(\begin{array}{ccc}-1 & 5 & 2 \\5 & -1 & 2 \\2 & 2 & 2 \end{array}\right)$

(d) $\left(\begin{array}{ccc}6 & -1 & 0 \\1 & 6 & -2 \\1 & 1 & 6 \end{array}\right)$

31) The sum of eigen values of $\left(\begin{array}{ccc}-1 & -2 & -1 \\-2 & 3 & 2 \\-1 & 2 & -3 \end{array}\right)$ is

(a) -3 (b) -1 (c) 3 (d) 1

32) The matrix $\left(\begin{array}{ccc}a^{2} & ab & ac \\ab & b^{2} & bc \\ac & bc & c^{2} \end{array}\right),$ where $a,b,c\in R-{ 0}$ has

(a) three real, non-zero eigen values (b) complex eigen values

33) $\left(\begin{array}{ccc}2 & 0 & 0 \\1 & 2 & 0 \\0 & 0 & 1 \end{array}\right)$ is

(a) diagonalizable (b) nilpotent (c) idempotent (d) not diagonalizable

34) If a square matrix of order 10 has exactly 5 distinct eigen values, then the degree of the minimal polynomial is

(a) at least 5 (b) at most 5 (c) always 5 (d) exactly 10

35) $T: R^{2\times 2}\rightarrow R^{2\times 2},$ defined by T(A)=BA, where $B=\left(\begin{array}{ccc}1 & -2 \\-2 & 4 \end{array}\right).$ Then rank of T is

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

36) $A=\left(\begin{array}{ccc}1 & a & b \\0 & 10 & c \\0 & 0 & 100 \end{array}\right).$ Then

(a) both $A$ and $A^{2}$ are diagonalizable (b) $A$ is diagonalizable but not $A^{2}$

37) Rank of $A_{7\times 5}$ is 5 and that of $B_{5\times 7}$ is 3, then rank of AB is

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

38) A and B are n-square positive definite matrices. Then which of the following are positive definite.

(a) A+B (b) ABA (c) AB (d) $A^{2}+I$

39) $A\in M_{3\times 3}(R)$ and $A=\left(\begin{array}{ccc}2 & 1 & 0 \\0 & 2 & 0 \\0 & 0 & 3 \end{array}\right),$ then which of the following are subspaces of $M_{3\times 3}(R)$

(a) ${X\in M_{3\times 3}(R): XA=AX}$ (b) ${X\in M_{3\times 3}(R):X+A=A+X}$

40) Let T be a linear operator on the vector space V and T be invariant under the subspace W of V. Then

(a) $T(W)\in W$ (b) $W\in T(W)$ (c) $T(W)=W$ (d) None of these

41) $A : R^{4}\rightarrow R^{3},$ where $A=\left(\begin{array}{cccc}1 & 2 & 3 & 1 \\1 & 3 & 5 & -2\\3 & 8 & 13 & -3 \end{array}\right).$ Then the dimension of kernel of A is

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

42) $A : R^{3}\rightarrow R^{4},$ where $A=\left(\begin{array}{ccc}1 & 1 & 3 \\2 & 3 & 8 \\3 & 5 & 13 \\1 & -2 & -3 \end{array}\right).$ Then the dimension of image of A is