4.4 The Determinant and the Inverse of a Matrix

The inverse of a square matrix M is a matrix, denoted as M^-1, with the property that M^-1 M = M M^-1 = I. Here I is the identity matrix of the same size as M, having 1's on the diagonal and 0's elsewhere.

In terms of transformations, M^-1 undoes the transformation produced by M and so the combination M^-1M represents the transformation that changes nothing.

The condition MM^-1 = I can be written as

and

when k and i are different, and these conditions completely determine the matrix M^-1 given M, when M has an inverse.

These equations have the same form as the two conditions (A) and (B) of section 4.3 except that det M is on the left-hand side in (A) instead of 1, and (-1)^{i + j}M_ij appears in (A) and (B) instead of M^-1_ji here.

We can therefore divide both sides of (A) and (B) by det M, and deduce

Remember that here M_ij is the determinant of the matrix obtained by omitting the i-th row and j-th column of M; the elements of M are the m_ij, while M^-1_ji here represents the element of the inverse matrix to M in j-th row and i-th column.

We can phrase this in words as: the inverse of a matrix M is the matrix of its cofactors, with rows and columns interchanged, divided by its determinant.

Exercises:

4.7 Compute the inverse of the matrix in Exercise 4.4 using this formula. Check the product M^-1M to be sure your result is correct.

4.8 Set up a spreadsheet that computes the inverse of any three by three matrix with non-zero determinant, using this formula.
(Hint: by copying the first two rows into a fourth and fifth row and the first two columns into a fourth and fifth column, you can make one entry and copy to get all of the (-1)^{i + j}M_ij at once. Then all that is left is rearranging to swap indices and dividing by the determinant (which is the dot product of any row of M with the corresponding cofactors).)