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# selected10.5 - Solutions to requested homework problems...

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Unformatted text preview: Solutions to requested homework problems from section 10.5 in the course notes: __________________________________________________________________ 0 3 5 4 1 0 2. Given A = , and B = , compute 3 A − 2 B. −2 3 −2 1 2 6 Solution: 0 9 15 8 2 0 3A = and 2 B = −4 6 −4 . By subtracting corresponding entries, we 3 6 18 9 − 2 15 − 0 −8 7 15 0−8 obtain 3 A − 2 B = = . 3 − ( −4) 6 − 6 18 − ( −4) 7 0 22 __________________________________________________________________ 5. Given 4 1 0 3 5 and C = 6 2 , compute AC − 3 I2 . A= 1 2 6 −2 3 Solution: 4 1 0 3 5 3 0 0 + 18 − 10 0 + 6 + 15 8 21 AC = 6 2 = 4 + 12 − 12 1 + 4 + 18 = 4 23 , and 3 I2 = 0 3 . 1 2 6 −2 3 8 − 3 21 − 0 5 21 Subtracting corresponding entries, we obtain AC − 3 I2 = . = 4 − 0 23 − 3 4 20 __________________________________________________________________ 1 1 1 9. Find the inverse of the matrix A = 3 2 −1 . 3 1 2 Solution: (The two column method below helps to reduce careless copying or arithemtic errors. ) Sequence of row equivalent matrices 1 1 1 1 0 0 −3r1 + r2 = R2 3 2 −1 0 1 0 → 3 1 2 0 0 1 “Scratch” −3r1 −3 −3 −3 −3 0 0 r2 3 2 −1 0 1 0 R2 0 − 1 −4 −3 1 0 Sequence of row equivalent matrices 1 1 1 1 0 0 −3r1 + r3 = R3 0 −1 −4 −3 1 0 → 3 1 2 0 0 1 1 1 0 0 1 1 r1 + r2 = R1 0 −1 −4 −3 1 0 → 0 −2 −1 −3 0 1 1 0 −3 −2 1 0 −2 r2 + r3 = R3 0 −1 −4 −3 1 0 → 0 −2 −1 −3 0 1 1 0 −3 −2 1 0 3r3 + rR1 = R1 0 −1 −4 −3 1 0 → 0 0 7 3 −2 1 7 0 0 −5 1 3 4 r3 + 7r2 = R2 0 −1 −4 −3 1 0 → 0 0 7 3 −2 1 “Scratch” −3r1 −3 −3 −3 −3 0 0 r3 3 1 2 001 0 −2 −1 −3 0 R3 r1 r2 11 1 100 0 −1 −4 −3 1 0 R1 1 0 −3 −2 1 −2r2 r3 R3 0 0 7 3 7 1 7 1 7 2 − 7 3 7 4 − 7 1 7 . -2 1 3r3 0 0 21 9 −6 3 7 r1 7 0 −21 −14 7 0 R1 70 0 −5 1 3 4r3 0 0 28 12 −8 4 7r2 0 −7 −28 −21 7 0 R2 0 −7 0 − 9 −1 4 7 0 0 −5 1 3 0 −7 0 −9 −1 4 → 0 0 7 3 −2 1 0 0 −5 7 9 10 7 0 13 0 0 2 8 6 −2 0 0 −2 −1 −3 0 1 1 r1 = R1 7 1 − r2 = R2 7 1 r3 = R3 7 2 1 0 0 1 − 5 7 9 So A−1 = 7 3 7 1 7 1 7 2 − 7 3 7 4 − . 7 1 7 ...
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## This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.

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