This verifies that I got the right answer for problem 3 on worksheet 29.
In Mathematica, ( ), { }, and [ ] all mean different things:
( ) is for grouping, { } is for lists including vectors and matrices, and [ ] is for the input to functions.  All three of these are used in this Mathematica notebook.

A = {{3, -13},{5,1}};  (* ";" suppresses output, like in MATLAB *)
MatrixForm[A]

Eigenvalues[A]

Solve x’ = Ax, x(0) = {3, -10} using the Swift method.  We find x1 = {133, 25}, so the solution to the IVP is

x[t_] = Exp[2t]( {3, -10} Cos[8t] + {133, 25} 1/8 Sin[8t]);

MatrixForm[x[t]]

It is clear that the solution satisfies the IC.  But does it really satisfy the ODE? The following should be the 0 vector.

x'[t] - A.x[t]

It does not look like the zero vector, so the “Simplify” command is helpful.

Simplify[x'[t] - A.x[t] ]