I used the following calculations in class on Feb 3, 2016.  These demonstrate some basic, but important Mathematica commands relevant to matrices.

a = {{-.08, .03}, {.08, -.08}};
MatrixForm[a]

CharacteristicPolynomial[a,x]

Roots[CharacteristicPolynomial[a,x]==0, x]

Eigenvalues[a]

Eigensystem[a]

JordanDecomposition[a]

MatrixExp[t a] // MatrixForm

Simplify[MatrixExp[t a].{{50},{25}}]

Here is another example.

b = {{0, -1},{1,0}};
MatrixForm[b]

CharacteristicPolynomial[b,z]

Eigenvalues[b]

Eigensystem[b]

JordanDecomposition[b]

MatrixExp[b t] // MatrixForm

MatrixPower[b, 5] // MatrixForm
b.b.b.b.b // MatrixForm