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The inverse of a square matrix is a matrix, denoted as , with the property that . Here is the identity matrix of the same size as , having 1's on the diagonal and 0's elsewhere.
In terms of transformations, undoes the transformation produced by and so the combination represents the transformation that changes nothing.
The condition can be written as
and
when and are different, and these conditions completely determine the matrix given , when has an inverse.
These equations have the same form as the two conditions (A) and (B) of section 4.3 except that is on the left-hand side in (A) instead of 1, and appears in (A) and (B) instead of here.
We can therefore divide both sides of (A) and (B) by , and deduce
Remember that here is the determinant of the matrix obtained by omitting the i-th row and j-th column of ; the elements of are the , while here represents the element of the inverse matrix to in j-th row and i-th column.
We can phrase this in words as: the inverse of a matrix 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 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
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
with the corresponding cofactors).)
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