Math254OldFinalSols

Math254OldFinalSols - FINAL - SOLUTIONS MATH 254 - SUMMER...

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FINAL - SOLUTIONS MATH 254 - SUMMER 2001 - KUNIYUKI 1) Find the inverse of the matrix A = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 040 100 002 . (8 points) Idea: AI IA [] Æ - 1 . 010 001 È Î Í Í Í ˘ ˚ ˙ ˙ ˙ È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 14 0 0 00 1 2 / / È Î Í Í Í ˘ ˚ ˙ ˙ ˙ RR 12 ´ 1 4 22 Æ 1 2 33 Æ So, A - = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 1 1 2 / / 2) Find the determinant of the matrix A = -- 000 3127 62 31 5004 2 . (10 points) Let's expand along the first row, since it has the most zeros. Its only nonzero entry is the " 2 "; the corresponding sign from the sign matrix is "-" [or you could observe that - () =- ++ 111 1 14 5 ij ]. To get the submatrix for the corresponding minor, we delete the row and the column containing the " 2 ". 0002 5 312 3 500 2 3 =- () ( ) -= - - from sign matrix {
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Let's expand along the third row, since it has the most zeros. Its only nonzero entry is the " 5 "; the corresponding sign from the sign matrix is "+" [or you could observe that - () =- () = ++ 111 1 31 4 ij ]. To get the submatrix for the corresponding minor, we delete the row and the column containing the " 5 ". -- = - - Ê Ë Á ˆ ¯ ˜ =- - - 2 312 62 3 2 5 12 23 253 4 257 70 500 3) Let v 1 = (1,2) and v 2 = (3,5). Express (-7,-9) as a linear combination of v 1 and v 2 . There should be no unknowns in your final expression. (10 points) Solve vv 7 9 - - È Î Í ˘ ˚ ˙ for c 1 and c 2 , the weights for the linear combination. 13 25 7 9 - - È Î Í ˘ ˚ ˙ 01 7 5 - - È Î Í ˘ ˚ ˙ 10 8 5 - È Î Í ˘ ˚ ˙ 8 5 - È Î Í ˘ ˚ ˙ RRR 212 2 121 3 RR 22 So, c 1 = 8 and c 2 = -5. - - È Î Í ˘ ˚ ˙ 7 9 85 4) The vectors x , y , and z are three vectors in R n for some fixed n .
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Math254OldFinalSols - FINAL - SOLUTIONS MATH 254 - SUMMER...

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