2003Fall20MidtermI

# 2003Fall20MidtermI - Math 20 Fall 2003 Midterm 1 1 Solve...

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Unformatted text preview: Math 20 Fall 2003 Midterm 1 1. Solve the following system of linear equations. x1 −2x1 3x1 − + − x2 2x2 x2 − + + 5x3 11x3 x3 Name = −1 =1 =3 2. Given that A = 3 −1 1 1 and B = 1 4 2 , ﬁnd (3AB )T − B T AT . 3 3. Given that (I − 3A)−1 = 2 4 0 , ﬁnd the matrix A. 1 1 4. Given that A = 0 2 4 0 10 6 1 1 and B = 0 9 0 46 0 1 , ﬁnd an elementary matrix E such that EA = B . 2 −3 1 5. Given that A = 0 0 2 2 0 1 2, ﬁnd A−1 . 4 r 6. Given that A = u x 00 10 (a) 0 −2 00 30 00 00 04 s v y t w and det(A) = 4, evaluate the following determinants. z 1 −2 3 −4 0 5 −6 7 (b) 0 0 −8 9 00 0 1 x (c) u r y v s z w t (d) r 8u x − 8r s 8v y − 8s t 8w z − 8t (e) det(A−1 ) (f) det(A2 ) 1 2 7. Evaluate the determinant 1 0 72 55 1 −1 00 0 0 . 0 2 8. Let A = 3 1 0 . 9 (a) Find the eigenvalues of the matrix A. (b) Select one of the eigenvalues you found in part (a) and ﬁnd its corresponding eigenvector. 9. Explain why the following statement is true. “If det(A) = 0, then the homogeneous system Ax = 0 has inﬁnitely many solutions.” 10. In each round of the two-person game rock-paper-scissors, each player chooses one of rock, paper, or scissors. Rock beats scissors, scissors beats paper, and paper beats rock. If both players choose the same object, then the round is considered a tie. (a) Give the payoﬀ matrix for this game, assuming that an entry of 1 indicates a win for the row player, an entry of −1 indicates a loss for the row player, and an entry of 0 indicates a tie. Label the rows and columns of the matrix with the row and column player actions. (b) If the row player chooses paper 50% of the time and scissors 50% of the time and if the column player chooses rock 75% of the time and scissors 25% of the time, which player will win more rounds of the game on average? ...
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2003Fall20MidtermI - Math 20 Fall 2003 Midterm 1 1 Solve...

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