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 ith column replaced by the vector b, Bi. Matrix matrix and matrix. Matrix matrix. Notice that the first column of A was replaced by the vector b. Replace the second column of A with b to get matrix. 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 Cramer's rule. Therefore, determinants, and determinants. 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

formula of the inverse or another formula of the inverse.


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, where

matrix.


We have already found that det(A) = -2, so let's find CT. Since we know how to transpose a matrix, let's start with finding C. The element

c_11 element, c_12 element, c_13 element,
c_21 element, c_22 element, c_23 element,
c_31 element, c_32 element, c_33 element.

Therefore,
determinants.
That means that
determinants, so

determinants.

This method is not used often because it requires that you find n2 determinants of dimension n - 1 by n - 1 and 1 determinant of dimension n by n. This means that this method would require approximately number of steps steps to compute the inverse if we computed the determinants with the best known algorithm. This is considerably more steps than are needed to compute the inverse using Gaussian elimination or Gauss-Jordan elimination.

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