The Adjoint Matrix and the Inverse

The adjoint of a matrix is defined to be a matrix with all of its elements replaced by their corresponding cofactors, transposed. Maple does this with the "adjoint" command, as follows:

> A:=matrix(6,6,rand(-5..5));

> adjoint(A);

For example, consider the (1,6) element of the above matrix, namely 6112. It should be the cofactor of the (6,1) element of A. Let's check this out.

Note that the (6,1) element is 5 steps away from the (1,1) element, which is an odd number of steps, so the sign will be negative:

> cofA61:=-det(minor(A,6,1));

This works. How about the (2,4) element of the adjoint matrix? It is 376. It should be the cofactor of the (4,2) element of A. Let's check this out.

Note that the (4,2) element is 4 steps away from the (1,1) element, which is an even number of steps, so the sign will be positive:

> cofA24:=det(minor(A,4,2));

Again, this works.

What use is the Adjoint Matrix? It can be used to compute the inverse of A, as follows:

> evalm(adjoint(A)/det(A));

Compare this to the direct computation of the inverse of A:

> inverse(A);

It is the same.

However, direct computation of the inverse using row reduction is much more efficient than computing the adjoint matrix.

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