Suppose we have one matrix $A=\left[\begin{array}{cc}4& 5\\ 8& 7\end{array}\right],$ and a vector which is $y=\left[\begin{array}{c}3\\ 4\end{array}\right],$ and we

would like to solve a linear system $Ax=y,$ where $x$ is an unknown vector. $(20`,5$

each$)$

$(1)$ Determine the value of $x$ using the standard inverse: $x=A^{-1}y$

$(2)$ Determine the value of tilde$(x)$ using the so$-$called pseudo inverse: tilde$(x)=(A^T(A){)}^{-1}{A}^{T}y,$

and show if $x$ equals to tilde$(x).$

$(3)$ For an arbitrary matrix A which is invertible, does $x$ always equal to tilde$(x),$ and why?

$($give the derivation$)$

$(4)$ Assume now $A=\left[\begin{array}{cc}4& 5\\ 8& 7\\ 2& 7\end{array}\right],$ and $y=\left[\begin{array}{c}3\\ 4\\ 2\end{array}\right],$ how can we calculate the vector $x?$