{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

winter 2007assign2

# winter 2007assign2 - An equivalent statement for the...

This preview shows page 1. Sign up to view the full content.

Math 235 Assignment 2 Due 9:15am, Wednesday Jan 24, 2007. 1. From the Text, § 3.1 #14. § 3.2 #10, #22. § 3 (Supplementary Exercises, page 211+) #14, #16. 2. Let A = [ a ij ] be an n × n matrix (over C , say). Let A ij denote the submatrix obtained from A by removing the i -th row and the j -th column, and let C ij be the cofactors of A defined by C ij = ( - 1) i + j det ( A ij ) . The matrix formed by the cofactors adj ( A ) = C 11 C 21 · · · C n 1 C 12 C 22 · · · C n 2 · · · · · · · · · · · · C 1 n C 2 n · · · C nn is called the classical adjoint of A . The fact that ( A )(adj ( A )) = det ( A ) I n is a theorem found in many text books. (cf. Theorem 8, page 203).
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: An equivalent statement for the theorem is that n ± j =1 a ij C kj = 0 for all i ± = k, (0.1) and n ± j =1 a ij C kj = det ( A ) for all i = k. (0.2) The latter is just the cofactor expansion for A along the i-th row. Give support for (0.1): For ﬁxed given i ± = k , exhibit a matrix B having two equal rows (or columns, thus det ( B ) = 0 is seen easily ) such that n ± j =1 a ij C kj (0.3) can be recognized as a cofactor expansion of B ....
View Full Document

{[ snackBarMessage ]}