DETERMINANTS

Suppose that A is an n × n matrix. One can define a quantity called the “determinant of A” for such a matrix A, often denoted by det(A). This handout summarizes some of the main properties of det(A).

Effect of elementary row operations on the determinant. Suppose that B is obtained from A by performing one elementary row operation. The effect on the determinant depends on the row operation as follows:

1. If B is obtained by adding a scalar multiple of one row of A to another row of A, then det(B)= det(A).

2. If B is obtained by multiplying one row of A by a nonzero scalar c, then det(B)= c·det(A).

3. If B is obtained by interchanging two rows of A, then det(B)= −det(A).

Notice that for each of the types of elementary row operations, det(B) = k · det(A), where k is nonzero.

Effect of matrix multiplication. Suppose that A and B are two n × n matrices. Then det(AB)= det(A) · det(B).

Triangular matrices. Suppose that A is an upper triangular matrix (or a lower triangular matrix). Then det(A) is equal to the product of the entries of A along the main diagonal. In particular, this applies to the special case where A is a diagonal matrix.

Nonsingular matrices. If A is a nonsingular matrix, then A is invertible. Let B = A−1. Then AB = In. Since In is a diagonal matrix with 1’s along its main diagonal, we have det(In) = 1. Therefore, we have det(AB) = 1. Since det(AB) = det(A)det(B), it follows that det(A)det(B) = 1. This fact implies that det(A) =6 0. It also follows that

−1 1 detA
= . det(A) If A is nonsingular, then A is row equivalent to In. That is, A can be transformed to In by a sequence of elementary row operations. Each elementary row operation can change the determinant, but only by multiplication by a nonzero scalar. Since det(In) = 1, it follows that det(A) =6 0. This is an alternative way of explaining why nonsingular matrices have nonzero determinants. If A is singular, then the reduced echelon form for A is a matrix E which has at least one row of zeros. It follows that det(E) = 0. Since A and E are row equivalent, it follows that det(E)= k · det(A), where k is a nonzero number (obtained from the combined effect of all the elementary row operations performed). Since det(E) = 0, it follows that det(A)=0. In summary, these arguments demonstrate the following very important fact concerning an n × n matrix A: A is nonsingular if and only if det(A) =6 0.

An interpretation of the determinant of A when n = 2 and n = 3. Suppose first of all that A is a 2 × 2 matrix and that v1 and v2 are the two columns of A. Consider the following set

P = { c1v1 + c2v2 : where 0 ≤ c1 ≤ 1, and 0 ≤ c2 ≤ 1 }

The set P is actually a parallelogram. It turns out that |det(A)| is the area of the parallelo- gram P.

Something similar is true for a 3 × 3 matrix A. Suppose that v1, v2, and v3 are the three columns of A. Consider the set

P = { c1v1 + c2v2 + c3v3 : where 0 ≤ c1 ≤ 1, 0 ≤ c2 ≤ 1, and 0 ≤ c3 ≤ 1 }

The set P is a parallelotope. It turns out that |det(A)| is the volume of the parallelotope P.