If A = (1 2 3), then the rank of AA^T is ________.

(a)

0

(b)

2

(c)

3

(d)

1

if T = \(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.4 } & \overset { B }{ 0.6 } \\ 0.2 & 0.8 \end{matrix} \right) \) is a transition probability matrix, then at equilibrium A is equal to ________.

(a)

\(\frac { 1 }{ 4 } \)

(b)

\(\frac { 1 }{ 5 } \)

(c)

\(\frac { 1 }{ 6 } \)

(d)

\(\frac { 1 }{ 8 } \)

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

(a)

all the minors of order r which does not vanish

(b)

A has at least one minor of order r which does not vanish

(c)

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

(d)

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

If the rank of the matrix  \(\left( \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right) \)  is 2. Then \(\lambda \) is ________.

(a)

1

(b)

2

(c)

3

(d)

only real number

If \(\rho (A)=\rho (A,B)\) then the system is ________.

(a)

Consistent and has infinitely many solutions

(b)

Consistent and has a unique solution

(c)

Consistent

(d)

inconsistent

If \(\rho(A) \neq \rho(A, B)\), then the system is _______.

(a)

Consistent and has infinitely many solutions

(b)

Consistent and has a unique solution

(c)

inconsistent

(d)

consistent

Rank of a null matrix is _______.

(a)

0

(b)

-1

(c)

\(\infty \)

(d)

1

For what value of k, the matrix \(A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right) \) has no inverse?

(a)

\(\frac { 3 }{ 10 } \)

(b)

\(\frac { 10 }{ 3 } \)

(c)

3

(d)

10

The value of \(\left| \begin{matrix} { 5 }^{ 2 } & { 5 }^{ 3 } & { 5 }^{ 4 } \\ { 5 }^{ 3 } & { 5 }^{ 4 } & { 5^{ 5 } } \\ { 5 }^{ 4 } & { 5 }^{ 5 } & { 5 }^{ 6 } \end{matrix} \right| \)

(a)

5²

(b)

0

(c)

5¹³

(d)

5⁹

If A is a singular matrix, then Adj A is ___________

(a)

non-singular

(b)

singular

(c)

symmetric

(d)

not defined