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

A scalar valued function is defined as $$f\left( x \right){x^T}Ax + {b^T}x + c,$$ where $$A$$ is a symmetric positive definite matrix with dimension $$n \times n;$$ $$b$$ and $$x$$ are vectors of dimension $$n \times 1$$. The minimum value of $$f(x)$$ will occur when $$x$$ equals.

For the matrix $$A$$ satisfying the equation given below, the eigen values are $$$\left[ A \right]\left[ {\matrix{ 1 & 2 & 3 \cr 7 & 8 & 9 \cr 4 & 5 & 6 \cr } } \right] = \left[ {\matrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 8 & 9 \cr } } \right]$$$

The dimension of the null space of the matrix $$\left[ {\matrix{ 0 & 1 & 1 \cr 1 & { - 1} & 0 \cr { - 1} & 0 & { - 1} \cr } } \right]$$ is