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**Tutu** 1.) If A is $\displaystyle \begin{bmatrix} 3 & 2 \\-2 & -1 \end{bmatrix}$, write A^2 in the form pA+qI where p and q are scalars. Hence write A^(-1) in the form rA+sI where r and s are scalars.

I know how to find A^2, I got $\displaystyle \begin{bmatrix} 5 & 4 \\-4 & -3 \end{bmatrix}$ but I do not know how to convert this matrix form into linear form pA+qI

2.) It is known that AB=A and BA=B where matrices A and B are not necessarily invertible.

Prove that A^2 = A.

When I first saw this, I thought B had to be I in AB=A and A in BA=B had to be I BUT they then added, NOTE: From AB=A, you cannot deduce that B=I. They asked me why, and I really dont know since I thought you could deduce that!

How do I then prove that A^2=A?