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NYC.L10 - MAT-NYC LAB#10 Invertible Matrices 1 Find the...

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MAT-NYC LAB #10 Invertible Matrices 1. Find the inverses of the following matrices (where possible): a) 1 0 - 1 2 - 1 0 1 1 1 b) 2 1 1 1 1 0 - 1 - 1 0 c) 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 d) 1 2 - 1 3 - 1 3 - 1 2 - 2 1 2 - 1 - 2 6 0 4 e) 5 - 7 - 2 3 , then solve for X : 3 X + 2 3 1 0 - 1 = 5 - 7 - 2 3 2. Let A = - 1 - 5 - 7 2 5 6 1 3 4 . Find the third column of A - 1 without computing the other columns. 3. Let A = 1 - 2 0 3 1 - 1 1 5 - 1 . a) Show that A is singular. b) Find a 3 × 3 matrix B such that B 6 = 0 and AB = 0. ( B is called a zero-divisor of A .) Hint : The columns of B must be in the nullspace of A . c) Find two 3 × 3 matrices M and N such that M 6 = N and AM = AN . Hint : Take N = M + B , where M is arbitrary and B is any zero-divisor of A . 4. Write the following system as a matrix equation A~x = ~ b , then solve it using inverses. x 1 + x 2 = k 1 where a) k 1 = 2 , k 2 = 0 , k 3 = 4 3 x 2 - x 3 = k 2 b) k 1 = 0 , k 2 = 4 , k 3 = - 2 x 1 + x 3 = k 3 c) k 1 = 4 , k 2 = 2 , k 3 = 0 5. Find the inverse of 1 - 1
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