5.3 Inverses and Solutions to Systems

Determinants also provide another way to solve the system Ax = b. The method we are going to describe is called Cramer's rule. We need one more bit of notation. We will call the matrix A with the i^th column replaced by the vector b, B_i. Let's use the example that we worked with in Chapter 4. Matrix and . Matrix . Notice that the first column of A was replaced by the vector b. Replace the second column of A with b to get . We need to find the determinants of each of these matrices. We find that |A| = 2, |B₁| = 12, and |B₂| = 42. The formula for Cramer's rule is . Therefore, , and . You should be happy to see that this is the same solution that Gauss-Jordan and Gaussian elimination gave us. Cramer's rule is essentially never used in computational mathematics because you are required to compute n + 1 determinants, where n is the dimension of the square matrix, before you can find your solution for x. This requires a lot more work than Gaussian elimination, so Cramer's rule is usually used only to examine theoretical properties of matrices. You can read more about this in the last chapter of this book.

There is another way to find the inverse. We can use the cofactors and determinants that we used when we expanded by minors. If we place all the cofactors into a matrix and call it C, the formula for the inverse is

or .

Notice that in the first formula, i and j are reversed on the opposite sides of the equation. Transposing matrix C yields the same result in the second equation. Let's find the inverse of the matrix A that we used in Chapter 4, where

.

We have already found that det(A) = -2, so let's find C^T. Since we know how to transpose a matrix, let's start with finding C. The element

, , ,
, , ,
, , .

.

, so

.