M313Lec09

M313Lec09 - Math 313 Lecture #9 2.1: The Determinant of a...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 313 Lecture #9 2.1: The Determinant of a Matrix Another Way to Determine Invertibility. Recall that an n n matrix is invertible if and only if A is row equivalent to I . Thus, one way to determine the invertibility of A is to row reduce it. There is another way to determine if A is invertible: it is possible to attach to A a scalar, det( A ), whose value determines whether A is invertible or not. We will start with the simplest square matrices and construct this scalar for larger square matrices. Case 1: 1 1 Matrices . If A = [ a ] is a 1 1 matrix, then A is invertible if and only if a 6 = 0, the inverse being A- 1 = [1 /a ]. Define the determinant of a 1 1 matrix A = [ a ] by det( A ) = a . Case 2: 2 2 Matrices . Let A = a 11 a 12 a 21 a 22 . Suppose that a 11 6 = 0. Then row reduction gives a 11 a 12 a 21 a 22 R 2- ( a 21 /a 11 ) R 1 R 2 a 11 a 12 a 22- a 12 a 21 /a 11 a 11 R 2 R 2 a 11 a 22 a 11 a 22- a 12 a 21 . Thus when a 11 6 = 0, the matrix A is row equivalent to I if and only if a 11 a 22- a 12 a 21 6 = 0, i.e., the determinant of a 1 1 submatrix is nonzero. Now, suppose that a 11 = 0. Then switching the rows of A gives a 21 a 22 a 12 which is row equivalent to I if and only if a 21 a 12 6 = 0 (neither entry on the diagonal can be zero). But a 21 a 12 6 = 0 is the same as a 11 a 22- a 12 a 21 6 = 0 when a 11 = 0, and so we arrive at the same expression as when a 11 6 = 0....
View Full Document

Page1 / 4

M313Lec09 - Math 313 Lecture #9 2.1: The Determinant of a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online