{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec3_3

lec3_3 - Example 5 Use cofactor expansion along the 3 rd...

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

NOTES ON SECTION 3 . 3 MA265, SECTIONS 41, 52 Key words minor expansion along a row/column cofactor area Definition 1. If A is a n × n matrix, let M ij be the matrix obtained by deleting the i th row and j th column. The ij th minor of A (or the minor of a ij ) is the number det M ij . Example 2. If A = 1 2 3 4 5 6 7 8 9 then M 23 = 1 2 7 8 and the minor of a 23 is det M 23 = 8 - 14 = - 6 . Definition 3. The cofactor of a ij is A ij = ( - 1) i + j det M ij . The cofactor is the minor of a ij up to a sign error, and the sign error is determined by the following “checkerboard” rule: + - + . . . - + - . . . + - + . . . . . . . . . . . . . . . Theorem 4. Pick a row r . Then det A = n X j =1 a rj A rj = a r 1 A r 1 + a r 2 A r 2 + · · · + a rn A rn Or, pick a column c . Then det A = n X i =1 a ic A ic = a 1 c A 1 c + a 2 c A 2 c + · · · + a nc A nc See the textbook/class notes for various examples of calculations using this method.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Example 5. Use cofactor expansion along the 3 rd row to calculate det A where A = 1-2 1 2 1 2-1-1 1 2 1 1 Since a 31 = a 32 = a 34 = 0 there is only one non-zero term det A = 0 + 0 + (-1) · A 33 + 0 where A 33 = (+) · det 1-2 2 1-1 1 2 1 1 Since it’s a 3 × 3 determinant we can compute this fairly easily to be 1 + 2 + 0--(-2)-(-4) = 9, so the ﬁnal answer is det A = a 33 A 33 = (-1) · 9 =-9 MATLAB agrees: >>> det([1,-2,1,0;2,1,2,-1;0,0,-1,0;1,2,1,1]) ans = -9 2...
View Full Document

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern