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32.3 The Inverse of a Matrix

If two square matrices M and A have the property that M A = I , (in infinite dimensions you also need the condition that A M = I ) then A and M are said to be inverses of one another and we write A = M − 1 and M = A − 1 .

A wonderful feature of row reduction as we have described it is that when you have a matrix equation A B = C , you can apply your reduction operations for the matrix A to the rows of A and C simultaneously and ignore B , and what you get will be as true as what you started with.

This is exactly what we did when B was the column vector with components equal to our unknowns, x , y and z , but it is equally true for any matrix B .

Thus, suppose you start with the matrix equation A A − 1 = I .

If we row reduce A so it becomes the identity matrix I , then the left hand side here becomes I A − 1 which is A − 1 , the matrix inverse to A . The right hand side however is what you obtain if you apply the row operations necessary to reduce A to the identity, starting with the identity matrix I .

We can conclude that the inverse matrix, A − 1 can be obtained by applying the row reduction operations that make A into I starting with I .

Example: we give a two dimensional example, but the method and idea hold in any dimension, and computers are capable of doing this for n by n matrices when n is in the hundreds even thousands.

Suppose we want the inverse of the following matrix

( 1 3 5 6 )

We can place an identity matrix next to it, and perform row operations simultaneously on both. Here we will first subtract 5 times the first row from the second row, then divide the second row by -9 then subtract three times the second from the first

( 1 3 5 6 ) A − 1 = ( 1 0 0 1 ) ( 1 3 0 − 9 ) A − 1 = ( 1 0 − 5 1 )

( 1 3 0 1 ) A − 1 = ( 1 3 5 / 9 − 1 / 9 ) ( 1 0 0 1 ) A − 1 = ( − 2 / 3 1 / 3 5 / 9 − 1 / 9 )

and the last matrix here is the inverse, A − 1 of our original matrix A .

Notice that the rows of A and the columns of A − 1 have dot products either 1 or 0 with one another, and the same statement holds with rows of A − 1 and columns of A . This is of course the defining property of being inverses.

Exercise 32.3 Find the inverse to the matrix B whose rows are first ( 2 4 ) ; second ( 1 3 ) . Solution

The inverse of a matrix can be useful for solving equations, when you need to solve the same equations with different right hand sides. It is overkill if you only want to solve the equations once.

If your original equations had the form M v ⟶ = r ⟶ , by multiplying both sides by M − 1 you obtain v ⟶ = I v ⟶ = M − 1 M v ⟶ = M − 1 r ⟶ , so you need only multiply the inverse, M − 1 of M by your right hand side, r ⟶ , to obtain a solution of your equations.

If you think of what you do here to compute the inverse matrix, and realize that in the process the different columns of M − 1 do not interact with one another at all, you are essentially solving the inhomogeneous equation M v ⟶ = r ⟶ for r ⟶ given by each of the three columns of the identity matrix, and arranging the results next to each other.

What we are saying here then is that to solve the equations for general r ⟶ it is sufficient to solve it for each of the columns of I , and then the solution for a general linear combination r ⟶ of these columns is the same linear combination of the corresponding solutions.

What matrices have inverses?

Not every matrix has an inverse.

As we have seen, when the rows of M are linearly dependent, the equations that M defines do not have unique solutions, which means that for some right hand sides there are lots of solutions and for some there are none. If so the matrix M does not have an inverse.

One way to characterize the linear dependence of the rows (or columns, if the rows are linearly dependent and the matrix is square, then the columns are linearly dependent as well) in three dimensions is that the volume of the parallelepiped formed by the rows (or columns) of M is zero.

The volume of the parallelepiped formed by the rows of M is not changed under the second kind of row operation, adding a multiple of a row to another, though it is changes by a factor | c | if you multiply each element of a row by c .

The fact that volume is always positive so that the absolute value | c | appears here is a bit awkward, and so it is customary to define a quantity that when positive is this volume but has the property of linearity: if you multiply a column by c it changes by a factor of c rather than by | c | . This quantity (and an analogue holds in any dimension) is called the determinant of M .

Thus the absolute value of the determinant of M is:

In one dimension the absolute value of the single matrix element of M .

In two dimensions the area of the parallelogram with sides given by the rows (or if you prefer, columns) of M .

In three dimensions the volume of the parallelepiped with sides given by the rows (or alternately the columns) of M .

In higher dimensions the "hypervolume" or higher dimensional analogue of volume of the region with sides given by the rows (or columns) of M .

In 0 dimensions we give whatever it is the value 1.

And the determinant of M in any dimension is linear in each of its rows or columns and is unchanged upon replacing one row , say q , by the sum of q and any multiple of any other row.

These statements specify the determinant up to a sign. The sign is determined by convention to be positive for the identity matrix I whose determinant is always 1.

The condition that M has an inverse is: the determinant of M is not zero.

We will soon see how to calculate determinants, and how to express the inverse of a matrix in terms of its determinant.

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