First we define a matrix to work with. To execute a comand-line place the cursor in the cell with a command, then hold Shift and press Enter (Shift+Enter).

mM = {{1, 2, 3}, {2, 3, 4}, {5, 6, 7}}

Now Mathematica knows what mM is:

MatrixForm[mM]

The next command line will calculate the characteristic polynomial.

Det[mM - λ IdentityMatrix[3]]

Eigenvalues[mM]

Eigensystem[mM]