Cramer's Rule and A Formula for Inverse
Cramer's Rule
Suppose is a matrix equation, where is an n x n matrix, and are n x 1 column vectors. Assuming is invertible, we can compute each of the entries in by the following formula:
,
where is the n x n matrix obtained from by replacing the ith column by the column vector :
This formula is called Cramer's rule. It is useful mainly for solving linear systems with variable coefficients.
Example: Consider the following system in which is an unspecified parameter. Determine the values of which the system has a unique solution, and use Cramer's rule to describe the solution.
Let . The above system can be expressed as the matrix equation , where and . Then , which means that the system has a unique solution exactly when .
To apply Cramer's rule, we need to compute and , where
Hence, we have
The proof of Cramer's rule:
Let be the n x n identity matrix, where is the standard basis. Suppose . We have
Taking determinants on both sides, we get
By a cofactor expansion along the ith row of , it is easy to see that . Hence, for .
A Formula for Inverse
We can easily derive a formula for from Cramer's rule. Let be the jth column of . Then . By Cramer's rule,
Notice that , where is the -cofactor of . Let be the adjugate of , the transpose of the matrix of cofactors:
Then the following is the formula for :
There is a nice youtube video here about using the above formula on a 3 x 3 matrix. However, we seldom use this formula in practice because it is much less efficient than using row operations. It is useful mainly for theoretical calculations.