Unit 4
Determinants I: Cofactor Expansion and Cramer’s Rule

In this unit, we introduce the concept of the determinant of a square matrix. The determinant is a function that associates a unique real number with any square matrix A. This number is denoted det(A), or |A|. The determinant of a matrix conveys some very important information.

If a and b are real numbers, then we know that the equation ax = b has the unique solution x = b/a, so long as a ≠ 0.

Another way of looking at this equation is to think of the coefficient a as a 1 × 1 matrix. Then the inverse, a-1 is just 1/a and we see that our solution is x = a-1b.

If a = 0, however, no solution exists, since we cannot divide by 0. The real number 0, considered as a matrix, has no inverse. In this unit, we will show that this relationship has an equivalent for any square matrix. That is, if A is any square matrix, then A-1 will exist, and hence Ax = b will have the unique solution x = A-1b if and only if det(A) ≠ 0.

In this unit, you will learn how to evaluate the determinant of a matrix by the method of cofactor expansion.

We also discuss how to find the cofactor matrix of a given square matrix, and based on this technique, the adjoint matrix. This process will lead to a formula for the inverse of an invertible matrix, which is of great theoretical importance.

Next, we consider Cramer’s Rule, a method for computing the solution to a system of n linear equations in n unknowns on the basis of the computation of n + 1 determinants. This method is particularly useful in computer applications, where programs for computing determinants are popular.

Objectives

When you have completed this unit, you should be able to

  1. compute the minor and the cofactor for any element of a square matrix.
  2. compute the determinant of a square matrix by cofactor expansion.
  3. define and compute the cofactor matrix and the adjoint matrix of any given square matrix.
  4. state and prove the formula for the inverse of a matrix in terms of its determinant and adjoint matrix.
  5. state and prove Cramer’s Rule, and use it to solve systems of linear equations.