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

If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________

(a)

a² + b² + c² = 1

(b)

abc ≠ 1

(c)

a + b + c =0

(d)

a² + b² + c² + 2abc =1

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

The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________

(a)

0

(b)

1

(c)

2

(d)

infinitely many

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

(a)

0

(b)

2n

(c)

2ⁿ

(d)

n²

The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________

(a)

k ≠ 0

(b)

-1 < k < 1

(c)

-2 < k < 2

(d)

k = 0

If A is a square matrix of order n, then |adj A| = ______________

(a)

|A|^n-1

(b)

|A|^n-2

(c)

|A|ⁿ

(d)

None

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 =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________

(a)

sinx cosx

(b)

1

(c)

2

(d)

none

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

If \(\rho\) (A) = r then which of the following is correct?

(a)

all the minors of order n which do not vanish

(b)

'A' has at least one minor of order r which does not vanish and all higher order minors vanish

(c)

'A' has at least one (r + 1) order minor which vanish

(d)

all (r + 1) and higher order minors should not vanish

Every homogeneous system ______

(a)

Is always consistent

(b)

Has only trivial solution

(c)

Has infinitely many solution

(d)

Need not be consistent

If \(\rho\) (A) ≠ \(\rho\) ([AIB]), then the system is _____________

(a)

consistent and has infinitely many solutions

(b)

consistent and has a unique solution

(c)

consistent

(d)

inconsistent

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

In a homogeneous system if \(\rho\) (A) =\(\rho\) ([A|0]) < the number of unknouns then the system has ________

(a)

trivial solution

(b)

only non - trivial solution

(c)

no solution

(d)

trivial solution and infinitely many non - trivial solutions

In the system of equations with 3 unknowns, if Δ = 0, and one of Δ_x, Δ_y of Δ_z is non zero then the system is ______

(a)

Consistent

(b)

inconsistent

(c)

consistent with one parameter family of solutions

(d)

consistent with two parameter family of solutions

In the system of liner equations with 3 unknowns If \(\rho\) (A) = \(\rho\) ([A|B]) =1, the system has ________

(a)

unique solution

(b)

inconsistent

(c)

consistent with 2 parameter -family of solution

(d)

consistent with one parameter family of solution.

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