(* TAYLOR EXPANSION AND MATRIX EXPONENTIAL *)

M:={{2,3},{3,2}}

M

MatrixForm[M]

M^2

M.M
MatrixPower[M,2]

sexp[s_,M_]:=(∑_(k=0)^s)((MatrixPower[M,k])/(k!))

sexp[5,M]

N[sexp[5,M]]

Round[45.8917]
Round[100 45.8917]/100.

Round[100N[sexp[5,M]]]/100.

MatrixForm[sexp[5,M]]
MatrixForm[N[sexp[5,M]]]
MatrixForm[Round[100N[sexp[5,M]]]/100.]

TableForm[Table[{s,MatrixForm[sexp[s,M]]},{s,0,20}]]

TableForm[Table[{s,MatrixForm[Round[100N[sexp[s,M]]]/100.]},{s,0,20}]]

M
M^2
Exp[M]

MatrixForm[MatrixExp[M]]

(****************************************)

(* COMPARNG NUMERICAL AND ANALYTICAL RESULTS *)

com[s_,M_]:={MatrixForm[M],s,
MatrixForm[Round[100N[sexp[s,M]]]/100.],
MatrixForm[Abs[Round[100N[sexp[s,M]-MatrixExp[M]]]/100.]],
MatrixForm[Round[100N[MatrixExp[M]]]/100. ],
MatrixForm[MatrixExp[M] ]}

com[8,M]

com[8,{{1,3},{3,1}}]

com[8,{{1,2},{2,1}}]