]> 4.4 The Determinant and the Inverse of a Matrix

## 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 $M − 1 M$ represents the transformation that changes nothing.

The condition $M M − 1 = I$ can be written as

$1 = ∑ j m i j M j i - 1$

and

$0 = ∑ j m k j M j i - 1$

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 $( − 1 ) i + j M i j$ appears in (A) and (B) instead of $M j i − 1$ here.

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

$M j i − 1 = ( − 1 ) i + j M i j det ⁡ M$

Remember that here $M i j$ 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 $m i j$ , while $M j i − 1$ 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 $M − 1 M$ 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 $( − 1 ) i + j M i j$ 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).)