Let $$A$$ be $$a$$ $$4$$ $$x$$ $$4$$ matrix with eigen values $$-5$$, $$-2, 1, 4$$.

Which of the following is an eigen value of $$\left[ {\matrix{ {\rm A} & {\rm I} \cr {\rm I} & {\rm A} \cr } } \right]$$, where $$I$$ is the $$4$$ $$x$$ $$4$$ identity matrix?

$$F$$ is an $$n$$ $$x$$ $$n$$ real matrix. $$b$$ is an $$n$$ $$x$$ $$1$$ real vector. Suppose there are two $$n$$ $$x$$ $$1$$ vectors, $$u$$ and $$v$$ such that $$u \ne v$$, and $$Fu = b,\,\,\,\,Fv = b$$

Which one of the following statements is false?

What are the eigen values of the matrix $$P$$ given below? $$$P = \left( {\matrix{ a & 1 & 0 \cr 1 & a & 1 \cr 0 & 1 & a \cr } } \right)$$$

Consider the following system of equations in three real variables $$x1, x2$$ and $$x3$$ :
$$2x1 - x2 + 3x3 = 1$$
$$3x1 + 2x2 + 5x3 = 2$$
$$ - x1 + 4x2 + x3 = 3$$
This system of equations has