Rank of a null matrix is _______.

\(\left| { A }_{ n\times n } \right| \) = 3 \(\left| adjA \right| \) = 243 then the value n is _______.

If \(\frac { { a }_{ 1 } }{ x } +\frac { { b }_{ 1 } }{ y } ={ c }_{ 1 },\frac { { a }_{ 2 } }{ x } +\frac { { b }_{ 2 } }{ y } ={ c }_{ 2 },\) \({ \triangle }_{ 1= }\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } \end{vmatrix}, \ { \triangle }_{ 2 }=\begin{vmatrix} { b }_{ 1 } & { c }_{ 1 } \\ { b }_{ 2 } & { c }_{ 2 } \end{vmatrix}{ \triangle }_{ 3 }=\begin{vmatrix} { c }_{ 1 } & { a }_{ 1 } \\ { c }_{ 2 } & a_{ 2 } \end{vmatrix}\) then (x, y) is _______.

Cramer’s rule is applicable only to get an unique solution when _______.

The system of linear equations x + y + z = 2, 2x + y − z = 3, 3x + 2y + k = 4 has unique solution, if k is not equal to _______.

If \(\left| A \right| \neq 0,\) then A is _______.

For the system of equations x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4 _______.

The system of equations 4x + 6y = 5, 6x + 9y = 7 has _______.

If the number of variables in a non-homogeneous system AX = B is n, then the system possesses a unique solution only when _______.

In a transition probability matrix, all the entries are greater than or equal to _______.

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

If \(\rho (A)=\rho (A,B)\) the number of unknowns, then the system is______.

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

Which of the following is not an elementary transformation?

If  \(T=\begin{array}{l} A \\ B \end{array}\left(\begin{array}{ll} 0.7 & 0.3 \\ 0.6 & x \end{array}\right)\) is a transition probability matrix, then the value of x is ________.

The rank of the diagonal matrix\(\left( \begin{matrix} 1 & & \\ & 2 & \\ & & -3 \end{matrix}\\ \quad \quad \quad \quad \quad \quad \quad \begin{matrix} 0 & & \\ & 0 & \\ & & 0 \end{matrix} \right) \)

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 ________.

If A =\(\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \) then the rank of AA^T is ________.

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

The rank of the unit matrix of order n is ________.

The rank of the matrix  \(\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix} \right) \) is ________.

If A = \(\begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}\),then \(\rho (A)\) is ________.

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 ________.

The rank of m x n matrix whose elements are unity is ________.

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