solnsExamI+2j - SOLUTIONS, EXAM I (1)(20 points) Answer...

Info iconThis preview shows pages 1–3. 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: SOLUTIONS, EXAM I (1)(20 points) Answer True or False for each statement and give a brief reason for your answer. ( a ) If A and B are n × n matrices and AB is nonsingular then both A and B are nonsingular. True: det( AB ) = det A det B, so det AB 6 = 0 if and only if det A 6 = 0 and det B 6 = 0 . Then use that A is non-singular if and only if det A 6 = 0 . ( b ) If the reduced row echelon form of the square matrix A has all zero entries in the last row, then A is nonsingular. False: counterexample 1 0 0 0 has all zeros in last row, its determinant is 0 and is therefore singular. ( c ) det a 11 + b 11 a 12 + b 12 . .. a 1 n + b 1 n a 21 a 22 . .. a 2 n . .. . .. a n 1 a n 2 . .. a nn = det a 11 a 12 . .. a 1 n a 21 a 22 . .. a 2 n . .. . .. a n 1 a n 2 . .. a nn + det b 11 b 12 . .. b 1 n a 21 a 22 . .. a 2 n . .. . .. a n 1 a n 2 . .. a nn 1 2 True: perform the determinant expansion using the first row, det a 11 + b 11 a 12 + b 12 . .. a 1 n + b 1 n a 21 a 22 . .. a 2 n . .. . .. a n 1 a n 2 . .. a nn = ∑ n j =1 ( a 1 j + b 1 j ) A 1 j = ∑ n j =1 a 1 j A 1 j + ∑ n j =1 b 1 j A 1 j = det...
View Full Document

This note was uploaded on 03/11/2012 for the course MATH 2J 44360 taught by Professor Donaldson,neil during the Spring '10 term at UC Irvine.

Page1 / 9

solnsExamI+2j - SOLUTIONS, EXAM I (1)(20 points) Answer...

This preview shows document pages 1 - 3. Sign up to view the full document.

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