Determinants (10/18/04) We learned previously the determinant of the 2 by 2 matrix A = is the number a d – b c. We need now to learn how to compute the determinant of larger square matrices. The definition is recursive, meaning we define the determinant of an n by n matrix in terms of the determinants of n different n – 1 by n – 1 matrices.

Definition of A i j and cofactors If A is an n by n matrix, then by A i j we mean the n – 1 by n – 1 matrix obtained by removing the i th row and the j th column of A. By the (i, j )-cofactor C i j of A, we mean the number (-1) i + j det(A i j ) The (-1) i + j causes the signs to alternate.

Definition of Det(A) If A is the n by n matrix [a i j ] with cofactors C i j, then for any fixed row i, det(A) = a i 1 C i 1 + a i 2 C i 2 +…+ a i n C i n One can also fix any column j. No matter which row or column you use, the same answer will emerge. Hence it makes sense to seek out a row or column with as many zeros as possible.

Some Special Cases If A has a row or column of all zeros, the its determinant must be zero A square matrix is called upper triangular if all the entries below the main diagonal are 0. Similarly lower triangular. The determinant of an upper or lower triangular matrix is just the product of the entries on the main diagonal.

Assignment for Wednesday Read Section 3.1 (Note: We shall return to some of the applications in Chapter 2, but no more of the regular material.) Do the Practice and Exercises 1 – 13 odd, 19, 21, and 23. In addition, explore the possible truth of the following two formulas (look at examples): det(A + B) = det(A) + det(B) ? Det(A B) = det(A) det(B) ?