MATH125 Lecture Notes - Lecture 27: Laplace Expansion, Main Diagonal, Triangular Matrix

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Nikita

Document Summary

The determinant of an n n matrix a = (aij) is the sum of n terms of the form a1j det a1j, with plus and minus alternating where the entries a11, a12 . , a1n are from the rst row of a. Thus det a = a11 det a11 a12 det a12 + + ( 1)1+na1n det a1n. 0 ] 5 det[ 2 1. We have det a = 1 det[ Instead of det we will also use vertical lines in place square brackets. For instance, instead of we also use the notation c d ] det[ a b (cid:12)(cid:12)(cid:12)(cid:12) c d (cid:12)(cid:12)(cid:12)(cid:12) a b. It is convenient to combine a minor with its plus or minus sign. The (i, j)-cofactor of a is the scalar. Cij = ( 1)i+j det aij, so that det a = a11c11 + a12c12 + + a1nc1n. This formula is called a cofactor expansion across the rst row of a.