If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

(a)

3

(b)

4

(c)

2

(d)

5

If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T = 

(a)

A

(b)

B

(c)

I₃

(d)

B^T

If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

(a)

-40

(b)

-80

(c)

-60

(d)

-20

If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

(a)

15

(b)

12

(c)

14

(d)

11

If A, B and C are invertible matrices of some order, then which one of the following is not true?

(a)

adj A = |A|A^-1

(b)

adj(AB) = (adj A)(adj B)

(c)

det A^-1 = (det A)^-1

(d)

(ABC)^-1 = C^-1B^-1A^-1

If A^TA⁻¹ is symmetric, then A² =

(a)

A^-1

(b)

(A^T)²

(c)

A^T

(d)

(A^-1)²

If x^a y^b = e^m, x^c y^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

(a)

e^{(Δ₂  / Δ₁)}, e^{(Δ₃ / Δ₁)}

(b)

log (Δ₁ / Δ₃), log (Δ₂ / Δ₃)

(c)

log (Δ₂ / Δ₁), log(Δ₃ / Δ₁)

(d)

e^{(Δ₁ / Δ₃)},e^{(Δ₂ / Δ₃)}

If \(\rho\) (A) = \(\rho\)([A| B]), then the system AX = B of linear equations is

(a)

consistent and has a unique solution

(b)

consistent

(c)

consistent and has infinitely many solution

(d)

inconsistent

If 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

(a)

\(\frac { 2\pi }{ 3 } \)

(b)

\(\frac { 3\pi }{ 4 } \)

(c)

\(\frac { 5\pi }{ 6 } \)

(d)

\(\frac { \pi }{ 4 } \)

The augmented matrix of a system of linear equations is \(\left[\begin{array}{cccc} 1 & 2 & 7 & 3 \\ 0 & 1 & 4 & 6 \\ 0 & 0 & \lambda-7 & \mu+5 \end{array}\right]\). The system has infinitely many solutions if

(a)

\(\lambda=7, \mu \neq-5\)

(b)

\(\lambda=-7, \mu=5\)

(c)

\(\lambda \neq 7, \mu \neq-5\)

(d)

\(\lambda=7, \mu=-5\)

The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________

(a)

λ = 8

(b)

λ = 8, μ ≠ 36

(c)

λ ≠ 8

(d)

none

Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________

(a)

±2

(b)

±4

(c)

±8

(d)

±12

If A^T is the transpose of a square matrix A, then ___________

(a)

|A| ≠ |A^T|

(b)

|A| = |A^T|

(c)

|A| + |A^T| =0

(d)

|A| = |A^T| only

If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______

(a)

0

(b)

2n

(c)

2ⁿ

(d)

n²

If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2 is inconsistent then k is ___________

(a)

-3, -\(\frac{1}{2}\)

(b)

-\(\frac{1}{2}\)

(c)

1

(d)

2

If A is a matrix of order m \(\times\) n, then \(\rho\) (A) is _________

(a)

m

(b)

n

(c)

≤ min (m,n)

(d)

≥ min (m,n)

The system of equations x + 2y + 3z = 1, x - y + 4z = 0, 2x + y + 7z = 1 has ___________

(a)

One solution

(b)

Two solution

(c)

No solution

(d)

Infinitely many solution

If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--

(a)

consistent and has infinitely many solutions

(b)

consistent

(c)

inconsistent

(d)

consistent and has unique solution

Which of the following is not an elementary transformation?

(a)

R_i ↔️ R_j

(b)

R_i ⟶ 2R_i + R_j

(c)

C_j ⟶ C_j + C_i

(d)

R_i ⟶ R_i + C_j

Every homogeneous system ______

(a)

Is always consistent

(b)

Has only trivial solution

(c)

Has infinitely many solution

(d)

Need not be consistent

In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______

(a)

unique solution

(b)

one parameter family of solution

(c)

two parameter family of solutions

(d)

in consistent

Cramer's rule is applicable only when ______

(a)

Δ ≠ 0

(b)

Δ = 0

(c)

Δ =0, Δ_x =0

(d)

Δ_x = Δ_y = Δ_z =0

If A = [2 0 1] then the rank of AA^T is ______

(a)

1

(b)

2

(c)

3

(d)

0

If A is a non-singular matrix then IA^-1| = ______

(a)

\(\left| \frac { 1 }{ { A }^{ 2 } } \right| \)

(b)

\(\frac { 1 }{ |A^{ 2 }| } \)

(c)

\(\left| \frac { 1 }{ A } \right| \)

(d)

\(\frac { 1 }{ |A| } \)

In a square matrix the minor M_ij and the co-factor A_ij of and element a_ij are related by _____

(a)

A_ij = -M_ij

(b)

A_ij = M_ij

(c)

A_ij = (-1)^i+j M_ij

(d)

A_ij =(-1)^i-j M_ij