Determinants Section The focus before was to determine information about the solution set of the linear system of equations given as the matrix equation Ax = b. We saw that in general, both the coefficient matrix A and right side b contributed to the specific nature of the solution set. This followed since the linear system Ax = b is consistent if and only if b is a linear combination of the columns of A. A goal was to determine the coefficients of such linear combinations and to do so we used row operations. However, in the special case of a nonsingular matrix A it was an intrinsic quality of the matrix A that the linear system Ax = b was consistent regardless of the contents of b. Here we develop an intrinsic number associated with a square matrix. This intrinsic number is called the determinant of the square matrix. We make a formal definition for the determinant of 2 2 and 3  3 matrices. We show how to generalize the determinant concept to n  n matrices. We develop properties of the determinant and show how it is related to linear systems that have a unique solution. We show how to use row operations to efficiently compute the determinant.

det(A) = numerical value The determinant of n × n matrix A, denoted det(A), is a function which associates with matrix A a unique numerical value. (The determinant is defined only for square matrices.) det(A) = numerical value

Computing the Determinant of 2 × 2 and 3 × 3 Matrices

Rather than compute det(A) using the expression given above we present the following scheme for generating the products required. Adjoin to A its first two columns, then compute the products along the lines shown in Figure 1. The products are then combined with the sign that appears at the end of the line. Those products obtained along the lines parallel to the diagonal have a plus sign associated with them and those not parallel to the diagonal have a minus sign. Figure 1. The computational procedure shown in Figure 1 is called the '3  3 device' for the determinant of a 3  3 matrix. Warning: there are no such short cuts or 'devices' for larger matrices.

Figure 2. det(A) = (-2) + (2) + - 0 - (-8) - (3) = 5

We have devices for computing the determinants of 2  2 and 3  3 matrices. How do we compute determinants of larger matrices? We discuss two techniques. The first technique expresses the determinant of a matrix A as a linear combination of determinants of submatrices of A. This called determinants by expansion. The second technique uses row operations on A to obtain an upper triangular matrix while we keep track of how the row operations used effect the value of the determinant.

Determinants by Expansion Here we develop a procedure for computing the determinant of an n × n matrix as a linear combination of determinants of submatrices matrices. We call this method determinants by expansion; it is also called Cofactor Expansion or Laplace Expansion by Cofactors. The expression for the determinant of a 3  3 matrix can be written as an expansion involving determinants of 2  2 submatrices and the entries from the first row. We have (verify by using the 2  2 device and simplifying) Note the use of the entries of row #1 as coefficients with signs attached.

To get the signs attached to the coefficients we use incorporate the row and column numbers of the rows and columns omitted by writing the expression in the form (1,1) entry = a (1,2) entry = b (1,3) entry = c We call this expression the expansion of the determinant about the first row. In fact, we can use any row or column for the expansion with appropriate powers of (-1) multiplying the entries and submatrices selected by omitting a row and column. Sign pattern for expansion method for a 3 × 3 matrix.

The definition we give for the determinant of an n  n matrix is an expansion about the first row following the pattern of the 3  3 case as given above. However, the expression involves determinants of (n -1)  (n -1) matrices, hence is recursive.

det(AB) = det(A) det(B). Suppose we have two matrices one is n × n matrix A and the other is n × n matrix B, we have det(AB) = det(A) det(B). Using magnification factor ideas we know that when A = identity matrix the magnification factor is 1 since the image is exactly the same as the original figure. Suppose A is nonsingular then we know A A-1 = I. Thus we have det(A A-1) = det(A) det(A-1) = det(I) = 1. For A nonsingular we have: Since a matrix is either singular or nonsingular we also have det(singular matrix) = 0.

Summary of the properties of the determinant

Computing the Determinant using Row Operations Computing determinant of an n × n matrix for large n using expansion is quite tedious. A more efficient technique involves the use of row operations. However, row operations can change the value of a determinant. The following properties can be proven; we will assume they are true.: and det(upper triangular matrix) = product of its diagonal entries. Using these four properties we will formulate a procedure for computing the determinant of any square matrix by using row operations to reduce it to upper triangular form and keeping track of how the row operations employed effect the determinant.

det of original matrix:

Another example: Compute the determinant of Ans = 184