Find Invertible and NonInvertible Matrix

Question

Can someone help me to understand this problem? I don't know where to begin.

Find an invertible matrix $A$ and a noninvertible matrix $B$ both of which satisfy $$M^2=3M$$

Thanks, Rusty

Answer 1

The comment gives you a very good way of constructing the simplest possible example: an example of size $1$.

Another simple kind of matrix that is helpful to look at is a diagonal matrix. For example, if we set $$ D = \pmatrix{d_1&0\\0&d_2} $$ then what values of $d_1,d_2$ make $D$ satisfy $D^2 = 3D$? From there, we could note that a matrix that is similar to $D$ would have the same property. That is, if $M = PDP^{-1}$ for some invertible matrix $P$, then $M^2 = 3M$.

Finally, you could try to classify all of the matrices satisfying the equation up to Jordan Canonical form. Once you're sufficiently familiar with this notion in linear algebra, you could prove that all matrices satisfying this equation must be similar to a matrix of the form $$ D = \pmatrix{d_1\\ &\ddots\\&&d_n} $$ where each $d_i$ is equal to either $0$ or $3$.