The following system of equations
$${x_1}\, + \,{x_2}\, + 2{x_3}\, = 1$$
$${x_1}\, + \,2 {x_2}\, + 3{x_3}\, = 2$$
$${x_1}\, + \,4{x_2}\, + a{x_3}\, = 4$$ has a unique solution. The only possible value (s) for $$\alpha $$ is/are

Let $$A$$ be the matrix $$\left[ {\matrix{ 3 & 1 \cr 1 & 2 \cr } } \right]$$. What is the maximum value of $${x^T}Ax$$ where the maximum is taken over all $$x$$ that are the unit eigenvectors of $$A$$?

The determination of the matrix given below is $$$\left[ {\matrix{ 0 & 1 & 0 & 2 \cr { - 1} & 1 & 1 & 3 \cr 0 & 0 & 0 & 1 \cr 1 & { - 2} & 0 & 1 \cr } } \right]$$$

What values of x, y and z satisfy the following system of linear equations? $$$\left[ {\matrix{ 1 & 2 & 3 \cr 1 & 3 & 4 \cr 2 & 3 & 3 \cr } } \right]\,\,\left[ {\matrix{ x \cr y \cr z \cr } } \right]\,\, = \,\left[ {\matrix{ 6 \cr 8 \cr {12} \cr } } \right]$$$