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

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

(c) rank of A is 1 or 0                   (d) A is diagonalizable

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

(c) $R^2,$ the zero subspace and the subspace with base {(1,1)}     (d) only $R^{2}$

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

(c)  both (a) and (b) are true  (d)   none of (a) and (b) is true

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

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

(c)  neither diagonalizable nor nilpotent   (d)   either diagonalizable or 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

(c)  both (a) and (b) are true   (d)   none of (a) and (b) is true

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

(c)  A and B are nilpotent   (d)  A and AB are 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

(c)   (2,-1,3), (0,1,1), (1,0,-2) are linearly dependent  (d)  S is a basis of $P_{2}(R)$

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

(c)   $T_{1}$ is onto and $T_{2}$ is 1-1   (d)   $T_{1}$ and $T_{2}$ both are 1-1

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

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

(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

(c)  A is idempotent   (d)   A is nilpotent

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

(c)  index of nilpotent is three   (d)   T is not nilpotent

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

(c)  A is nilpotent   (d)   minimal polynomial ≠ characteristics polynomial

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

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

(c)  both (a) and (b) are true   (d)   none of (a) and (b) is true

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

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

(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

(c)  two non-zero eigen values   (d)  only one non-zero eigen value

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 $A^{2}$

(c)   $A$ and $A^{2}$ have the same minimal polynomial  (d)   $A^{2}$ is diagonalizable but not $A$

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)   ${X\in M_{3\times 3}(R):X+A=A+X}$

(c)   ${X\in M_{3\times 3}(R): trace(AX)=0}$   (d)   ${X\in M_{3\times 3}(R): det(AX)=0}$

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

(a)  1  (b)  2  (c)  3   (d)   4

43) Let u, v, w be three non-zero vectors which are linearly independent, then

(a)  u is linear combination of v and w  (b)  v is linear combination of u and w

(c)  w is linear combination of u and v   (d)   none of these

44) Let U and W be subspaces of a vector space V and $U\cup W$ is also a subspace of V, then

(a)  either $U\subseteq W$ or $W\subseteq U$    (b)   $U\cap W= \phi$   (c)  U=W   (d)   None of these

45) Let I be the identity transformation of the finite dimensional vector space V, then the nullity of I is

(a)  dimV  (b)  0  (c)   1  (d)   dimV – 1

46) $T : R^{3}\rightarrow R^{3}$  such that $T(a,b,c)=(0, a,b),$ for $(a,b,c)\in R^{3}.$ Then $T+I$ is a zero of the polynomial:

(a)   $t$  (b)   $t^{2}$  (c)   $t^{3}$  (d)   none of above

47) The sum of the eigen values of the matrix   $\left(\begin{array}{ccc}4 & 7 & 11 \\7 & 1 & -21\\11 & -21 & 6\end{array}\right)$ is

(a)  4  (b)  23  (c)  11   (d)   12

48) Let A and B are square matrices such that AB=I, then zero is an eigen value of

(a)  A but not of B  (b)  B but not of A  (c)  both A and B   (d)  neither A nor B

49) The eigen values of a skew-symmetric matrix are

(a)  negative  (b)  real  (c)  absolute value of 1   (d)   purely imaginary or zero

50) The characteristics equation of a matrix A is $t^{2}-t-1=0,$ then

(a)   $A^{-1}$ does not exist (b)   $A^{-1}$ exit but cannot be determined from the data

(c)   $A^{-1}=A+1$   (d)   $A^{-1}=A-1$

The answers can be found here.