"Some simple Matrix operations in Mathematica"
"Defining a n by m matrix"

M1={{a11,a12,a13},{a21,a22,a23},{a31,a32,a33}}

"MatrixForm[M1]"

"Dimenstions of a matrix"
Dimensions[M1]

"Matrix multiplication "

M2={{a11,a12},{a21,a22}}
M3={{b11,b12},{b21,b22}}
"M2.M3="
M2.M3

"MatrixForm[M2.M3]="
MatrixForm[M2.M3]

"Inverse of a Matrix: Inverse[M2] = "
Inverse[M2]

"Matrix Form a an inverse, MatrixForm[Inverse[M2]] = "
MatrixForm[Inverse[M2]]

(({{(a22/(-a12 a21+a11 a22)), -(a12/(-a12 a21+a11 a22))}, {-(a21/(-a12 a21+a11 a22)), (a11/(-a12 a21+a11 a22))}}))

"Multiply a matrix with its inverse must get back an identity matrix"
"M2.Inverse[M2]="
M2.Inverse[M2]

"MatrixForm[M2.Inverse[M2]]"
MatrixForm[M2.Inverse[M2]]

"Simplify[M2.Inverse[M2]]="
Simplify[M2.Inverse[M2]]