Section 6.3 Geometrical Interpretations of the Determinant; Cramer's Rule

Interpertation 1 - Area, Volume, and m-Volume

The determinant of an orthogonal matrix is either 1 or -1.

An orthogonal n x n matrix A with det(A) = 1 is called a rotation matrix, and the linear transformation T(x) = Ax is called a rotation.

Consider a 2 x 2 matrix A = [ v1 v2]. The area of the parallelogram defined by v1 and v2 is |det(A)|.

If A is an n x n matrix with columns v1, v2, ..., vn, then |det(A)| = ||v1|| ||v2perp|| ... ||vnperp||, where v2perp, ..., vnperp are defined by the Gram-Schmidt process.

Consider a 3 x 3 matrix A = [ v1 v2 v3]. Tthe volume of the parallelpiped defined by v1, v2, and v3 is |det(A)|.

Generalization - Consider the vectors v1, v2, ..., vm in R^n. The m-volume of the m-parallelpiped defined by the vectors v1, v2, ..., vm is sqrt(det(A^TA)), where A is the n x m matrix with column v1, v2, ..., vm. If n = m, then the m-volume of the m-parallelpiped is |det(A)|. See p. 278 for the recursive definition of the m-volume or an m-parallelpiped in R^n.

Interpretation 2 - Expansion Factor

Consider a linear transformation T(x) = Ax from R^2 to R^2. Then |det(A)| is equal to the

expansion factor = (area of T(omega)) / (area of (omega))

Cramer's Rule

If a matrix is invertible, this formula allows us to express the inverse in terms of its determinant.

Consider the linear system Ax = b, where A is an invertible n x n matrix. The components xi of the solution vector x are xi = det(A_b,i) / det(A), where A_b,i is the matrix obtained by replacing the ith column of A by b.

Group/Class Exercises

Section 6.3 - 1, 2, 3, 4, 9, 10, 12, 13, 18, 23, 24, 25

Please complete the exercises above for class on Tuesday.