The figure shows a shape $$ABC$$ and its minor image $${A_1}$$$${B_1}$$$${C_1}$$ across the horizontal axis ($$x$$-axis). The coordinate transformation matrix that maps $$ABC$$ to $${A_1}$$$${B_1}$$$${C_1}$$ is

The eigen values of the matrix $$A = \left[ {\matrix{ 1 & { - 1} & 5 \cr 0 & 5 & 6 \cr 0 & { - 6} & 5 \cr } } \right]$$ are

If $$V$$ is a non-zero vector of dimension $$3 \times 1,$$ then the matrix $$\,A = V{V^T}$$ has a rank $$=$$ ________

Let $$A$$ be an $$n \times n$$ matrix with rank $$r\left( {0 < r < n} \right).$$ Then $$AX=0$$ has $$p$$ independent solutions, where $$p$$ is