Exercise 8.9.2

Determine which of the following matrices are diagonalizable.

1. $A$ is a $3\times 3$-matrix with eigenvalues $-1$ and $3$. The eigenvalue $-1$ has algebraic multiplicity $2$ and geometric multiplicity $1$. The eigenvalue $3$ has algebraic and geometric multiplicity $1$.

2. $B$ is a $4\times 4$-matrix with eigenvalues $2$ and $-2$. The eigenvalue $2$ has algebraic and geometric multiplicity $1$. The eigenvalue $-2$ has algebraic and geometric multiplicity $3$.

3. $C$ is a $5\times 5$-matrix with eigenvalues $1$ and $3$. The eigenvalue $1$ has algebraic and geometric multiplicity $2$, and the eigenvalue $3$ has algebraic and geometric multiplicity $1$.