Math 22 Fall 2004

Linear Algebra with Applications

The Inverse of a Matrix

October 11, 2004

    Load the packages for doing  Linear Algebra

>	with(Student[LinearAlgebra]):

Warning, the protected name `.` has been redefined and unprotected

   Define a matrix we want to invert

>	A := <<0, 3, -1>|<2, -2, 0>|<1, -5, 1>>;

   Augment it with the identical matrix

>	A_I := <A | IdentityMatrix(3)>;

    Perform row reductions to transform A into

    the Reduced Echelon Form (that is, the identity matrix).

>	A_I := SwapRows(A_I, 1, 3);

>	A_I := AddRow(A_I, 2, 1, 3);

>	A_I := AddRow(A_I, 3, 2, 1);

>	A_I := MultiplyRow(A_I, 3, -1): A_I := MultiplyRow(A_I, 2, -1/2): A_I := MultiplyRow(A_I, 1, -1);

>	A_I := AddRow(A_I, 2, 3, -1): A_I := AddRow(A_I, 1, 3, 1);

    Now the last 3 columns should form the inverse matrix of A

>	A_Inverse := LinearAlgebra[DeleteColumn](A_I, [1 .. 3]);

      Let's check our answer

>	A.A_Inverse, A_Inverse.A;

>