SOLUTION: {{{A=(matrix(2,2,2,-5,3,1))}}} Evaluate the square of A and find x,y,z, not all zero, such that the matrix xI+yA+zAA is the zero matrix.

We will find those three terms separately:





To find zAA we first find the square of A, which is AA



(That was the first thing asked for, the square of A)

Now we multiply that by z to get zAA:



Next we find their sum:



This must equal to the zero 2×2 matrix, so,



We form this system by setting each element = 0:



We solve that system by Gaussian elimination:



By using row operations we find the row reduced echelon form
of that augmented matrix.



The first two rows of that matrix gives us
the system:



or



To find all solutions, we choose a constant k for z, then



Edwin