heorem 3.1. For all A 2 Rn n we have det(A = det A>). Proof. For any pattern P of A we there is a corresponding pattern P> of A> obtained in the obvi us fashion. The numerical entries of P and …

Determinant and elementary row operations Theorem 3 If E represents an elementary row operation and A is an n n matrix, then det(EA) = det(E) det(A): 5 The proof is to compute the …

The determinant is a number associated with any square matrix; we’ll write it as det A or |A|. The determinant encodes a lot of information about the matrix; the matrix is invertible exactly when …

Corollary 4.6 (Inverses). A is invertible (non-singular) if and only if det A−1 = 1 det A and 1 c A

det(A) = det(E1) det(E2) · · · det(Em). Recall there are easy formulas for the determinnats and inverses of elementary matrices. However, it is generally more efficient to work with the row …

det(A) and |A| It turns out that it is more natural to think of the determinant as a function whose input is n vectors from Rn – e.g. the row vectors of an outputs a real number. The essential …

Theorem 3.2.1: Product Theorem If A and B are n × n matrices, then det (AB) = det A det B. The complexity of matrix multiplication makes the product theorem quite unexpected. Here is an …