**1) If A and B are symmetric matrices of same order and X = AB + BA and Y = AB – BA, then (XY) ^{T} is equal**

(a) **XY ** (b) **YX ** (c) **-YX** (d) **none of these**

(c) X = AB + BA ⇒ X^{T} = X

and Y = AB – BA ⇒ Y^{T} = -Y

Now, (XT)^{T} = Y^{T} × X^{T} = -YX.

**2) If A is a skew–symmetric matrix of order 3, then the matrix A ^{4} is **

(a) **skew symm****etric** (b) **symmetric** (c) **diagonal** (d) **none of these**

(b) We have A^{T} = –A

(A^{4})T = (A.A.A.A.)^{T} = A^{T} A^{T} A^{T} A^{T}

⇒ [–(A)][–(A)][–(A)][–(A)]

= (–1)^{4} A^{4} = A^{4 }

Q3.If** A = i -i and B = 1 -1**

[ **-i ** **i **] [**-1** **1 **] **, then A ^{8} equals**

(a) **4B** (b) **128B** (c) **-128B** (d) **-64B**

(B) We have A = iB

⇒ A^{2} = [i(B)]^{2} = i^{2}B^{2} = -B^{2} = – 2 -2 = -2B

[-2 2]

⇒ A^{4} = [-2(B)]^{2} = 4B^{2} = 4[2(B)] = 8B

⇒ (A^{4})^{2} = [8(B)]^{2}

⇒ A^{8} = 64B^{2} = 128B

**4) A square matrix A is said to be nilpotent of index m. If A ^{m} = 0, now, if for this A [I – (A)]^{n} = I + A + A^{2} + … + A^{m-1}, then n is equal to **

(a) **0** (b) **m** (c) **-m** (d) **-1**

(d) Let B = I + A + A^{2} + … + A^{m-1}

⇒ B (I – (A) = (I + A + A^{2} + … + A^{m-1}) [I – (A)] = I – A^{m} = I

⇒ B = [I – (A)]^{-1}

⇒ n = -1.

**5) If A and B are symmetric matrices then AB – BA is a**

(a) **symmetric matrix** (b) **skew symmetric matrix**

(c) **diagonal matrix** (d) **null matrix.**

(b) A’ = A , B’ = B

(AB – BA)’ = (AB)’- (BA)’ = B’A’- A’B’

= BA – AB

= – (AB – BA) = skew symmetric

**6) If A and B are square matrices of same order & A is nonsingular then for a +ve integer n, [A ^{–1} B(A)]^{n} is equal to **

(a) **A ^{–n} B^{n} A^{n} ** (b)

**A**(c)

^{n}B^{n}A^{–n }**A**(d)

^{–1}B^{n}A**n[A**

^{–1}B(A)](c) [A^{–1} B(A)]^{2} = [A^{–1}B(A)] = [A^{–1}B(A)]

= A^{–1}B^{2}A

Sum [A^{–1} B(A)]^{n} = A^{–1}B^{n}A.

**7) Let A and B be two 2 × 2 matrices. Consider the statements **

**(i) AB = O ⇒ A = O or B = O **

**(ii) AB = I _{2} ⇒ A = B^{-1} **

**(iii) (A + (B) ^{2} = A^{2} + 2AB + B^{2} then**

^{ }(a) **(i) is false, (ii) and (iii) are true** (b) **(i) and (iii) are false, (ii) is true**

(c) **(i) and (ii) are false, (iii) is true** (d) **(ii) and (iii) are false, (i) is true **

(C) [A^{–1} B(A)]^{2} = [A^{–1}B(A)] = [A^{–1}B(A)]

= A^{–1}B^{2}A

Sum [A^{–1} B(A)]^{n} = A^{–1}B^{n}A.

**Q8.If A = [aij] _{3 × 3}, such that a_{ij} = when i = j , **

** {when i ≠ j , then 1 + log _{1/2}det[Adj(Adj (A)] = **

(A) -12 (B) -10 (C) -13 (D) -11

(d) A = [a_{ij}]

a_{ij} = 2, i = j = 0, i ≠ j

|A| = 8

Now, det [Adj(Adj(A)] = (8)^{4} = 2^{12}

**9) A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(a) **(3 ^{9}-3^{6}-3^{3}+1)/3^{9}** (b)

**(3**(c)

^{9}-3^{6}-3^{3})/3^{9 }**(3**(d)

^{9}-1)/3^{10}**(3**

^{9}-3^{3}+1)/3^{9}(a) Total number of matrices that can be formed is 3^{9} .

Let A = [a_{ij}]_{3×3 }where a_{ij} ∈ { 1,0,1}

If A is symmetric then a_{ij} = a_{ji} ∀ i , j

If A is skew-symmetric then a_{ij} = -a_{ji} ∀ i , j

**Q10.A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(A) 1/abc(ab+bc+ca) (B) ab+bc+ca^{ } (C) 0 (D) a+b+c

(C) Given determinant = b^{2}c^{2}(ca^{2} – a^{2}b)+c^{2}a^{2}(ab^{2} – b^{2}c)+a^{2}b^{2}(bc^{2} – c^{2}a)

**Q11.A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(A) 0 (B) 1^{ } (C) 100 (D) -100

(C) Given determinant = b^{2}c^{2}(ca^{2} – a^{2}b)+c^{2}a^{2}(ab^{2} – b^{2}c)+a^{2}b^{2}(bc^{2} – c^{2}a)

**Q12.A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(A) 31/15 (B) 41/18^{ } (C) 50/21 (D) 61/27

(C) Given determinant = b^{2}c^{2}(ca^{2} – a^{2}b)+c^{2}a^{2}(ab^{2} – b^{2}c)+a^{2}b^{2}(bc^{2} – c^{2}a)

**Q13.A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(A) -5 (B) -4^{ } (C) -3 (D) -2

^{2}c^{2}(ca^{2} – a^{2}b)+c^{2}a^{2}(ab^{2} – b^{2}c)+a^{2}b^{2}(bc^{2} – c^{2}a)

**Q14.A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(A) 5π/4 (B) -3π/4^{ } (C) π/4 (D) -π/4

^{2}c^{2}(ca^{2} – a^{2}b)+c^{2}a^{2}(ab^{2} – b^{2}c)+a^{2}b^{2}(bc^{2} – c^{2}a)

**Q15.A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(A) 5π/4 (B) -3π/4^{ } (C) π/4 (D) -π/4

(d) pqr(a3+b3+c3) – abc(p3+q3+r3)

⇒pqr(a3b3c3 – 3abc) – abc(p3+q3+r3-3pqr)

⇒pqr(a3+b3+c3 – 3abc) – abc(p+q+r)(p2+q2+r2 – pq – qr – rp)

= pqr(a3+b3+c3 – abc).

**Q16.A is a 3×3 matrix with entries from the set { -1,0,1} . The probability that A is neither symmetric nor skew symmetric is**

(A) (3n – r) (B) 2(3n – r)^{ } (C) 3(3n – r) (D) 3n + r

(b) LHS = (24mx+6n) – (12mx2+nx+2r)

x = 0 ⇒ 6n – 2r ⇒ 2(3n – r)

**17) ****The number of values of k for which the system of equations kx+(k+3)y = 3k – 1; (k+1)x+8y = 4k**** has infinitely many solutions is **

(a) **0** (b) **1**^{ } (c) **2 ** (d) **infinite**

(b) Solving the equations we get

(k – 1)(k – 3)y = -(k -1)^{2}

If k = 1 , infinite sol.

If k ≠ 1 then inconsistent if k = 3

If k ≠ 1,3, then kx = [4k(k -2)]/(k -3)

If k = 0 unique sol.

If k ≠ 0 unique sol.

**18) Given 2x-y+2z = 2 , x-2y+z = 4 , x+y+λz = 4 then the value of λ such that the given system of equation has no solution is**** **

(a) **3** (b) **1**^{ } (c) **0** (d) **-3**

(B) Solving the equations we get

(k – 1)(k – 3)y = -(k -1)^{2}

If k = 1 , infinite sol.

If k ≠ 1 then inconsistent if k = 3

If k ≠ 1,3, then kx = [4k(k -2)]/(k -3)

If k = 0 unique sol.

If k ≠ 0 unique sol.

**19) If the system of linear equations x+y+z = 6 , x+2y+3z = 14 and 2x+5y+λz = μ , (λ , μ ∈ R) has no solution , then**

(a) **λ ≠ 8** (b) **λ = 8, μ ≠ 36**^{ } (c) **λ = 8, μ = 36** (d) **None of these**

(B) Solving the equations we get

(k – 1)(k – 3)y = -(k -1)^{2}

If k = 1 , infinite sol.

If k ≠ 1 then inconsistent if k = 3

If k ≠ 1,3, then kx = [4k(k -2)]/(k -3)

If k = 0 unique sol.

If k ≠ 0 unique sol.

**Q20.If the system of linear equations x+y+z = 6 , x+2y+3z = 14 and 2x+5y+λz = μ , (λ , μ ∈ R) has no solution , then**

(A) 3ω (B) 3ω(ω – 1) (C) 3ω^{2} (D) 3ω(1 – ω)

(B) Solving the equations we get

(k – 1)(k – 3)y = -(k -1)^{2}

If k = 1 , infinite sol.

If k ≠ 1 then inconsistent if k = 3

If k ≠ 1,3, then kx = [4k(k -2)]/(k -3)

If k = 0 unique sol.

If k ≠ 0 unique sol.

**Q21.If the system of linear equations x+y+z = 6 , x+2y+3z = 14 and 2x+5y+λz = μ , (λ , μ ∈ R) has no solution , then**

(A) -11,6 (B) -6,11 (C) 6,11 (D) -6,-11

(B) Solving the equations we get

(k – 1)(k – 3)y = -(k -1)^{2}

If k = 1 , infinite sol.

If k ≠ 1 then inconsistent if k = 3

If k ≠ 1,3, then kx = [4k(k -2)]/(k -3)

If k = 0 unique sol.

If k ≠ 0 unique sol.

**Q22.If the system of linear equations x+y+z = 6 , x+2y+3z = 14 and 2x+5y+λz = μ , (λ , μ ∈ R) has no solution , then**

(A) -11,6 (B) -6,11 (C) 6,11 (D) -6,-11

(a) If Q = PAP^{T}

P^{T}Q = AP^{T} (1)(as PP^{T} = 1)

P^{T}Q^{2005}P = AP^{T}Q^{2004}P

= A^{2}P^{T}Q^{2003}P

= A^{3}P^{T}Q^{2002}P

= A^{2004}P^{T}(QP)

= A^{2004}P^{T}(PA) (Q = PAP^{T} ⇒ QP = PA)

= A^{2005}

= A^{2005}