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Question Bank No: 1

1. The value of the determinant $[\begin{matrix} 1 & ω^{6} & ω^{8} \\ ω^{6} & ω^{3} & ω^{7} \\ ω^{8} & ω^{7} & 1 \end{matrix}]$ ,where $ω^{2}$ =1, is

a)		3
b)		– 3
c)		$(1 - ω)^{2}$
d)		None of these

2. For a square matrix A and a non-singular matrix B of the same order, value of determinant of $B^{- 1}$ AB is

a)		$\| A \|$
b)		$\| B \|$
c)		$\| B^{- 1} \|$
d)		$\| A^{- 1} \|$

3. If p + q + r = a + b + c, then value of the determinat $[\begin{matrix} pa & qb & rc \\ qc & ra & pb \\ rb & pc & qa \end{matrix}]$ is

a)		0
b)		pa +qb + rc
c)		1
d)		None of these

4. If x, y, z are in A.P., then the value of the determinant $[\begin{matrix} a + 2 & a + 3 & a + 2 x \\ a + 3 & a + 4 & a + 2 y \\ a + 4 & a + 5 & a + 2 z \end{matrix}]$ is

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

5. The value of a for which the system of equations: ax + y + z = 0, x + ay + z = 0, x + y + z = 0 possesses non-null solution is

a)		1
b)		1, 2
c)		1, -1
d)		None of these

6. If for a square matrix A, $A^{2}$ = A then $| A |$ is equal to

a)		0 or 1
b)		- 2 or 2
c)		- 3 or 3
d)		None of these

7. If A = $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ , then $A^{2}$ + 5A is

a)		A
b)		2A
c)		5A
d)		6A

8. The multiplicative inverse of A = $[\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}]$ is equal to

a)		A
b)		A'
c)		$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$
d)		$[\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}]$

9. For a 2 $\times$ 2 matrix A, if A (adj. A) = $[\begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix}]$ , then $| A |$ is

a)		0
b)		10
c)		20
d)		100

10. If A is an invertible matrix, then det ( $A^{- 1}$ ) is equal to

a)		1
b)		$\| A \|$
c)		$\frac{1}{\| A \|}$
d)		None of these

11. If a matrix A is symmetric as well as skew symmetric, then

a)		A is a diagonal matrix
b)		A is a nul matrix
c)		A is a unit matrix
d)		A is a triangular matrix

12. The number of all possible matrices of order 3 $\times$ 3 with each entry 0 or 1 is

a)		18
b)		512
c)		81
d)		None of these

13. A and B be 3 $\times$ 3 matrices. Then AB = O implies

a)		A = O and B = O
b)		$\| A \|$ = 0 and $\| B \|$ = 0
c)		either $\| A \|$ = 0 or $\| B \|$ = 0
d)		A = O or B = O

14. If A = $[a_{ij}]_{m \times n}$ is a skew-symmetric matrix, then $a_{ii}$ is

a)		0 for each i
b)		0 for some i
c)		1 for each i
d)		1 for some I

15. If A is skew-symmetric then $A^{n}$ for an odd positive integer is

a)		Symmetric
b)		skew-symmetric
c)		diagonal
d)		None of these

16. If A is symmetric then $A^{n}$ for n $\in$ N is

a)		Symmetric
b)		skew-symmetric
c)		diagonal
d)		scalar

17. If A, B are two square matrices such that AB = A and BA = B then

a)		A, B are indempotent
b)		only A is indempotent
c)		only B is indempotent
d)		None of these

18. If A is an orthogonal matrix, then $A^{- 1}$ is

a)		A
b)		A'
c)		$A^{2}$
d)		None of these

19. If A and B are two matrices such that AB = B and BA = A then $A^{2}$ + $B^{2}$ is equal to

a)		2AB
b)		2BA
c)		A + B
d)		AB

20. If A is an m $\times$ n matrix such that AB and BA are both defined, then B is an

a)		m $\times$ n matrix
b)		n $\times$ m matrix
c)		n $\times$ n matrix
d)		m $\times$ m matrix

21. If A= $[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] and n \in N then A^{n} is equal to$

a)		$2^{n} A$
b)		$2^{2 - 1} A$
c)		nA
d)		none of these

22. If B is a non singular matrix and A is a square matrix then det( $B^{- 1} AB) is equal to$

a)		det(B)
b)		det(A)
c)		det( $B^{- 1})$
d)		det( $A^{- 1})$

23. If AB=A and BA=B then $B^{2} is equal to$

a)		A
b)		B
c)		1
d)		none of these

24. If A= $[\begin{matrix} 1 & 2 & - 1 \\ 1 & 1 & 2 \\ 2 & - 1 & 1 \end{matrix}] then$

a)		f(A)=1
b)		f(A)=2
c)		f(A)=3
d)		none of these

25. The system of linear equation ax+by=0,cx+dy=0 has non trivial solution if

a)		ad-bc $> 0$
b)		ad-bc $< 0$
c)		ac+bd=0
d)		ad-bc=0

26. For any 2 $\times$ 2 matrix A, if A(adjA)= $[\begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix}] then | A | =$

a)		0
b)		10
c)		20
d)		100

27. If f(x)= $[\begin{matrix} 1 & x & x + 1 \\ 2 x & (x - 1) x & (x + 1) x \\ 3 x (x - 1) & x (x - 1) (x - 1) & (x + 1) x (x - 1) \end{matrix}] then f (100) is$

a)		0
b)		1
c)		100
d)		-100

28. A square matrix can always be expressed as a

a)		sum of a symmetric matrix and a skew symmetric matrix
b)		sum of a diagonal matrix and a symmetric matrix
c)		skew matrix
d)		skew symmetric matrix

29. For a square matrix A, it is given that A $A^{1} = I then A is a$

a)		orthogonal matrix
b)		diagonal matrix
c)		symmetric matrix
d)		none of these

30. The matrix $[\begin{matrix} 0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0 \end{matrix}] is known as$

a)		symmetric matrix
b)		upper triangular matrix
c)		diagonal matrix
d)		skew symmetric matrix

31. $Δ = | \begin{matrix} 1 & 1 + ac & 1 + bc \\ 1 & 1 + ad & 1 + bd \\ 1 & 1 + ae & 1 + be \end{matrix} |$

a)		1
b)		0
c)		3
d)		a+b+c

32. If $[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}] the cofactor of a_{r} is A_{r} then C_{1} A_{1} + C_{2} A_{2} + C_{3} A_{3} is$

a)		0
b)		-D
c)		D
d)		$D^{2}$

33. If A and B are square matrix of the same type then

a)		A+B=B+A
b)		A+B=A-B
c)		A-B=B-A
d)		AB=BA

34. If a matrix A is such that 3 $A^{3} + 2 A^{2} + 5 A + I = 0 then its matrix is$

a)		-(3 $A^{2} + 2 A + 5 I)$
b)		3 $A^{2} + 2 A + 5 I$
c)		3 $A^{2} - 2 A - 5 I$
d)		none of these

35. If each element of a 3 $\times$ 3 matrix is multiplied by 3, then the determinant of the newly formed matrix is

a)		3 det A
b)		9 det A
c)		27 det A
d)		81det A

36. If In is the identity of order n then ${In}^{- 1} is$

a)		does not exist
b)		In
c)		0
d)		nIn

37. If A= $[\begin{matrix} 3 & 6 \\ 2 & 4 \end{matrix}] then A is$

a)		singular
b)		non singular
c)		1
d)		$A^{- 1} = [\begin{matrix} 4 & - 2 \\ - 6 & 3 \end{matrix}]$

38. If A= $[\begin{matrix} a & b \\ c & d \end{matrix}] \underset{..}{s} ad - bc \neq 0 then A^{- 1} is$

a)		$\frac{1}{ad - bc} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$
b)		$[\begin{matrix} d & - b \\ - c & a \end{matrix}]$
c)		$\frac{1}{ad - bc} [\begin{matrix} d & b \\ - c & a \end{matrix}]$
d)		none of these

39. If A= $[\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}] then adj A is equal to$

a)		$[\begin{matrix} \frac{1}{2} & - 2 \\ - 2 & 1 \end{matrix}]$
b)		$[\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}]$
c)		$[\begin{matrix} 1 & - 2 \\ - 2 & - 1 \end{matrix}]$
d)		$[\begin{matrix} - 1 & 2 \\ 2 & - 1 \end{matrix}]$

40. If $A^{1} is the transpose of a square matrix A then$

a)		$\| A \| \neq \| A^{1} \|$
b)		$\| A \| = \| A^{1} \| for all A$
c)		$\| A \| + \| A^{1} \| = 0$
d)		$\| A \| = \| A^{1} \| only$

41. If a square matrix A has a column of zeros, then the determinant of A is

a)		0
b)		1
c)		-1
d)		none of these

42. If A,B,C be three square matrixes such that A=B+C then determinent A is equal to

a)		det B+det C
b)		det B
c)		det C
d)		none of these

43. If A and B are square matrix of order 3 S $| A | = - 1 | B | = 3 then the determinent of 3 AB is equal to$

a)		-9
b)		-27
c)		-81
d)		81

44. If the capital letters denote cofactor of the corresponding small letters in the determinent $Δ = | \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} | Δ^{1} = | \begin{matrix} A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3} \end{matrix} | is$

a)		$Δ$
b)		$Δ^{2}$
c)		2 $Δ$
d)		0

45. The value of $Δ = | \begin{matrix} 1 & 1 & 1 \\ 1 & w & w^{2} \\ 1 & w^{2} & w \end{matrix} | is$

a)		3 $\sqrt{3} i$
b)		-3 $\sqrt{3} i$
c)		- $\sqrt{3} i$
d)		$\sqrt{3} i$

46. If $Δ = | \begin{matrix} 5 & 15 & - 25 \\ 7 & 21 & 30 \\ 8 & 24 & 42 \end{matrix} | then$

a)		$Δ = 0$
b)		$Δ = 1$
c)		$Δ = - 1$
d)		$Δ = 5$

47. The value of the determinent $| \begin{matrix} a + b + 2 c & a & b \\ c & b + c + 2 a & b \\ c & a & c + a + 2 b \end{matrix} |$ is

a)		2(a+b+c)
b)		2 ${(a + b + c)}^{3}$
c)		ab+bc+ca
d)		2abc(ab+bc+ca)

48. If A+B+C= $\prod then the value of | \begin{matrix} \sin (A + B + C) & sinB & cosC \\ - sinB & 0 & tanA \\ \cos (A + B) & - tanA & 0 \end{matrix} | is$

a)		0
b)		1
c)		2sinB tanA tanC
d)		none of these

49. If $Δ = | \begin{matrix} 27 & 40 & 58 \\ 24 & 36 & 52 \\ 18 & 28 & 40 \end{matrix} |$

a)		$Δ = 0$
b)		$Δ = 1$
c)		$Δ = 2$
d)		$Δ = - 2$

50. If $Δ = | \begin{matrix} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{matrix} | then Δ is equal to$

a)		4
b)		1
c)		0
d)		2