Matrices and Determinants

The number of A in $\mathrm{T}_p$ such that $\operatorname{det}(\mathrm{A})$ is not divisible by $p$ is :

The number of A in $\mathrm{T}_p$ such that the trace of A is not divisible by $p$ but $\operatorname{det}(\mathrm{A})$ is divisible by $p$ is
[Note : The trace of a matrix is the sum of its diagonal entries.]

The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A}) \operatorname{divisible}$ by $p$ is :

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solu...

The number of matrices A in A for which the system of linear equations \(A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]\) is inconsistent, is

The number of matrices A in A for which the system of linear equations \(A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]\) has a unique solution, is

The number of matrices in A is

Consider the system of equations:
\(x-2y+3z=-1\)
\(-x+y-2z=k\)
\(x-3y+4z=1\)
Statement - 1 : The system of equations has no solution for \(k\ne3\).
and
Statement - 2 : The determinant $$\left| {\matrix{
1 & 3 & { - 1} \cr
{ - 1} & {...

The sum of the elements of $\mathrm{U}^{-1}$ is:

The value of $\left[\begin{array}{lll}3 & 2 & 0\end{array}\right] U\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is :

The value of \(|U|\) is :

If \(\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0\) and the vectors
$$\overrightarrow A = \left( {1,a,{a^2}} \right),\,\,\overrightarrow B = \...