For which values of $a$ is the matrix $A=([7-a,-15],[2,-4-a])$ invertible?

In the following linear system, determine all values of a for which the resulting linear syste

a$)$ a unique solution

b$)$ no solutions

c$)$ infinitely many solutions

$x+y-z=2$

$x+2y+z=3$

$x+y+(a^2-5)z=a$

Consider the following vectors: $a=([-1],[2],[-2]),b=([x],[y],[1]).$ Find the value of $x$ and $y$ when you $k$ that the length of their cross product is $3\sqrt[\phantom{2}]{5},$ and vectors are perpendicular to each other.

Solve for the matrix A$,$ then find the null space and the rank of A$.$

$(5A^T+([1,0],[2,5]){)}^T=3A+([1,-2],[-1,3]{)}^{-1}$