Section 5.4 Examples
Subsection 5.4.1 Inverses of matrices
There is a general formula for the inverse of a formula:
The formula in the denominator of the scalar multiple in this inverse formula is called the determinant of Clearly the formula does not work when the determinant of is since we cannot divide by zero. In fact, in Chapter 6 it will be possible for us to prove that is not invertible when There are similar formulas for inverses of larger matrices, but they are too complicated to write down explicitly. We will study the general theory of determinants and related inversion formulas in Chapters 8–10.
Example 5.4.1. Using the inversion formula.
Let’s check that we have the correct inverse. To keep the computations simple, we’ll leave the as a scalar multiple when expressing
Example 5.4.2. Sometimes the inversion formula does not apply.
Consider matrix
For this matrix, we have
Subsection 5.4.2 Solving systems using inverses
Just as we can solve the numerical equation as we can solve a system of equations that is represented by a matrix equation as in cases where the coefficient matrix is square and invertible.
Example 5.4.3.
Consider the system
The coefficient matrix for this system is
which is conveniently the matrix for which we have already computed the inverse using the inversion formula in Subsection 5.4.1. So we can solve the system as
Subsection 5.4.3 Solving other matrix equations using inverses
We can similarly use matrix algebra and inverses to solve matrix equations in general.
Example 5.4.4.
Consider the matrix equation
One approach to this problem would be to express in terms of unknown entries,
and then set up four equations in the four unknowns This would lead to a system of equations that we could row reduce and solve. But it’s easier just to use ordinary (matrix) algebra. Set
substitute these definitions into the given equation, and isolate algebraically:
Of course, this method wouldn’t work if was not invertible, but it is, and we can calculate
From this we obtain