# HW5 - AMS210.01. Homework 5 Andrei Antonenko Due at the...

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Unformatted text preview: AMS210.01. Homework 5 Andrei Antonenko Due at the beginning of the class, April 14 1. Determine the number of inversions and the sign of the following permutations: (a) (51423) (b) (54321) (c) (n n − 1 n − 2 . . . 2 1) (all numbers from 1 to n in the reverse order) 2. Check which of the following terms are included in the expression for the determinant (for 5 × 5-matrix) and with which signs. (a) a15 a21 a35 a42 a53 (b) a13 a25 a31 a45 a54 (c) a22 a34 a41 a12 a55 3. Compute the following determinants: (a) 35 53 (b) ab ac bd cd 4. Compute the following determinants: 123 (a) 5 1 4 325 0a0 (b) b c d 0e0 5. Compute the following determinants: a11 0 0 ... 0 a21 a22 0 . . . 0 (a) a31 a32 a33 . . . 0 ........................ an1 an2 an3 . . . ann 1 0 ... 0 0 a1n 0 ... 0 a2,n−1 a2n (b) 0 . . . a3,n−2 a3,n−1 a3n .............................. an1 . . . an,n−2 an,n−1 ann 6. How do the determinant of the matrix changes if (a) change the sign of all entries of the matrix. (b) to each of the rows add all the rows preceding it. (c) put the ﬁrst row on the last place, and all other rows move up. 7. Compute the following determinants using elementary row operations. 1 10 100 1000 10000 100000 0.1 2 30 400 5000 60000 0 0.1 3 60 1000 15000 (a) . 0 0 0.1 4 100 2000 0 0 0 0.1 5 150 0 0 0 0 0.1 6 1 2 3 ... n −1 0 3 ... n (b) −1 −2 0 . . . n ..................... −1 −2 −3 . . . 0 1 n n ... n n 2 n ... n (c) n n 3 . . . n ................ n n n ... n 1 a1 a2 ... an 1 a1 + b1 a2 ... an (d) 1 a1 a2 + b2 . . . an ................................. 1 a1 a2 . . . an + bn 8. Using the expansion by the 3rd row, ﬁnd the determinant: 2 −3 4 −2 ab 3 −1 2 4 3 c 4 1 2 d 3 9. Compute the following determinants using expansion by a row (or column) x y 0 ... 0 0 0 x y ... 0 0 0 0 x ... 0 0 (a) .................... 0 0 0 ... x y y 0 0 ... 0 x a0 −1 0 0 ... 0 0 a1 x −1 0 . . . 0 0 a2 0 x −1 . . . 0 0 (b) ................................. an−1 0 0 0 . . . x −1 an 0 0 0 ... 0 x 10. Compute the following determinant by squaring the matrix. a b c d −b a d −c −c −d a b −d c −b a 11. Solve the following systems by Cramer’s rule (a) 2x1 + 5x2 = 1 3x1 + 7x2 = 2 x1 + x2 + x3 = 6 (b) −x1 + x2 + x3 = 0 x1 − x2 + x3 = 2 (c) x1 cos α + x2 sin α = cos β −x1 sin α + x2 cos α = sin β 12. [Extra credit] It is known that numbers by 17. Prove that the determinant 2 5 2 2 0 20604, 53227, 25755, 20927 and 289 can be divided 0 3 5 0 0 divides by 17. 3 6 2 7 9 2 0 2 5 2 8 4 7 5 7 9 13. [Extra credit] Compute the following determinant (use properties!) a1 + x a2 ... an a1 a2 + x . . . an ............................ a1 a2 . . . an + x 14. [Extra credit] Compute the following determinant (use elementary row operations). x1 a12 a13 . . . a1n x1 x2 a23 . . . a2n x1 x2 x3 . . . a3n ....................... x1 x2 x3 . . . xn 15. [Extra credit] Compute the following determinant (use elementary row operations). 1 1 ... 1 −n 1 1 . . . −n 1 ....................... 1 −n . . . 1 1 −n 1 . . . 1 1 16. [Extra credit] Compute the following determinant (use expansion). a0 1 1 1 . . . 1 1 a1 0 0 . . . 0 1 0 a2 0 . . . 0 ....................... 1 0 0 0 . . . an 17. [Extra credit] Compute the following determinant (use expansion). a1 0 ... 0 b1 0 a2 ... b2 0 ............................ 0 b2n−1 . . . a2n−1 0 b2n 0 ... 0 a2n 18. [Extra credit] Compute the following determinant (use recurrency). 0 1 1 ... 1 1 1 0 x ... x x 1 x 0 ... x x ................... 1 x x ... 0 x 1 x x ... x 0 4 ...
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