Math 375 - 11/13

Question 1 • A=(■8(1&1&a@−1&1&b@0&2&c)) • For which a,b,c∈R is A invertible? • When A is invertible, find A^(−1) • Answer: ○ det⁡A=−2a−2b+2c ○ cof A=(■8(c−2b&c&−2@2a−c&c&−2@b−a&−a−b&2)) ○ adj A=(cof A)^T=(■8(c−2b&2a−c&b−a@c&c&−a−b@−2&−2&2)) ○ A^(−1)=1/(−2a−2b+c) (■8(c−2b&2a−c&b−a@c&c&−a−b@−2&−2&2)) ○ Where a+b≠c Question 2 • Let A be square matrix such that A^k=0 for some k • Prove or find a counterexample : I−A is invertible • Answer: ○ I=I−A^{k=(I−A)(I+A+A}2+…A^(k−1) ) ○ Therefore I−A is invertible • Note: ○ A is called Nilpotent matrix