## Linear Algebra and Its Applications, exercise 1.3.8

Exercise 1.3.8. Given a system of equations of order n = 600, how long would it take to solve in terms of the number of multiplication-subtractions? In seconds, on a PC capable of 8,000 operations per second? On a VAX system capable of 80,000 operations per second? On a Cray X-MP/2 capable of 12 million operations per second?

Answer: As discussed in section 1.3, subsection The Cost of Elimination, for large n the number of operations is approximately $\frac{1}{3}n^3$. For n = 600 the number of operations is therefore approximately $\frac{1}{3} \cdot 600^3 = \frac{1}{3} \cdot 6^3 \cdot 100^3 = \frac{1}{3} \cdot 2^3 \cdot 3^3 \cdot (10^2)^3 = 2^3 \cdot 3^2 \cdot 10^6 = 8 \cdot 9 \cdot 1,000,000 = 72,000,000$

At 8,000 operations per second this would take 72,000,000 / 8,000 = 9,000 seconds (two and a half hours). At 80,000 operations per second this would take 900 seconds (15 minutes), At 12 million operations per second this would take six seconds.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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