3.4 Inverses

In-class demo by Russell Blyth.

> restart: with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

We start with a random 3 x 3 matrix.

> A:= randmatrix(3,3,entries=rand(-3..3));

Find the inverse of the matrix A. The equation Y = AX computes Y given X. We need to compute X given Y. Define a general Y, and solve for X:

> Y := vector(3,[y1,y2,y3]);
AugA := augment(A,Y);
RedA := gaussjord(AugA);

Let's extract the expression for X in terms of Y

> Xvec := delcols(RedA,1..3);

> y1vec := subs(y1=1,y2=0,y3=0,op(Xvec));
y2vec := subs(y1=0,y2=1,y3=0,op(Xvec));
y3vec := subs(y1=0,y2=0,y3=1,op(Xvec));

Thus Xvec = y1*y1vec + y2*y2vec + y3*y3vec. If we make a matrix with y1vec, y2vec, y3vec as the columns, then y1, y2 and y3 are the coefficients of the linear combination giving X. We write this as BY, where

> B := augment(y1vec,y2vec,y3vec);

Check:

> evalm(B&*Y);

This is just the transpose of Xvec. Thus X = BY. B is the inverse of A!

Check the products AB and BA:

> evalm(A&*B);
evalm(B&*A);

Note the first column of B arises from solving AX = ; the second column from solving AX = and the third column from solving AX = . These three systems may be solved simultaneously, by finding the reduced row echelon form of the matrix [A,I]

> I3 := array(identity,1..3,1..3);
BigA := augment(A,I3);
RedBigA := gaussjord(BigA);

Note the final three columns are precisely the inverse matrix B.

Maple has a built-in inverse command:

> inverse(A);

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