solMidTerm2A - Solutions to Midterm #2, MAT1341A Question #...

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Unformatted text preview: Solutions to Midterm #2, MAT1341A Question # 1: (3 points) Let A = 1 1 3 and B = 2 4 1 1 2 3 1 3 5 . Compute AB T . Solution: AB T = 1 1 3 1 2 3 1 1 = 2 2 2 3 6 9 Question # 2: (2+2+2 = 6 points) The following augmented matrices are in RREF. Write down the general solution of the corresponding linear system. (a) 2 6 6 4 1 3 1 1 2 3 7 7 5 Solution: This system has a unique solution, since it is consistent and there is a leading one in every column. We read o the solution from the matrix; note that there are exactly 3 variables, one for each column, so the general solution is f (3 ; ; 2) g (b) 1 Solution: This question is similar to one given on HW5. It is a matrix in RREF, so we proceed as usual. There are 4 variables but only x 2 corresponds to a leading 1; it gives equation x 2 = 0. The rest are non-leading variables, so correspond to parameters. We get x 1 = r; x 2 = 0 ; x 3 = s; x 4 = t r; s; t 2 R so the general solution is f ( r; ; s; t ) j r; s; t 2 R g . (c) 2 4 1 1 3 1 1 2 3 3 5 Solution: This system is inconsistent, because the last line corresponds to a degenerate equation. Therefore the general solution, which is by de nition the set of all solutions to the system, is the empty set (denoted ; or fg ). Question # 3: (5 points) Reduce the following matrix to RREF, using Gaussian elimination. 2 6 6 4 1 2 2 1 1 2 4 1 1 1 3 3 3 1 2 1 4 3 7 7 5 1 2 Solution: 2 6 6 4 1 2 2 1 1 2 4 1 1 1 3 3 3 1 2 1 4 3 7 7 5 R 2 ! R 2 +2 R 1 ! R 4 ! R 4 R 1 2 6 6 4 1 2 2 1 1 3 3 3 3 3 3 3 1 3 3 7 7 5 ! R 2 ! 1 3 R 2 2 6 6 4 1 2 2 1 1 1 1 1 3 3 3 3 1 3 3 7 7 5 R 3 ! R 3 +3 R 2 ! R 4 ! R 4 +3 R 2 2 6 6 4 1 2 2 1 1 1 1 1 2 6 3 7 7 5 R 3 R 4 ! R 3 ! 1 2 R 3 2 6 6 4 1 2 2 1 1 1 1 1 1 3 3 7 7 5 R 2 ! R 2 R 3 ! R 1 ! R 1 R 3 2 6 6 4 1 2 2 2 1 2 1 3 3 7 7 5 ! R 1 ! R 1 2 R 2 2 6 6 4 1 2 2 1 2 1 3 3 7 7 5 Recall that RREF means reduced row echelon form | the best form for writing down the solution to a linear system; it is the optimal form in the sense of having the most zeros. Do notice that you should follow a systematic procedure to get to RREF (although shortcuts are ne once you know what you're doing!). Putting zeros \at random" in the matrix will not achieve your goal....
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solMidTerm2A - Solutions to Midterm #2, MAT1341A Question #...

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