Chapter 2.4, Problem 45E

To gauge the complexity of a computational task, mathematicians and computer scientists count the number ofelementary operations (additions, subtractions, multiplications, and divisions) required. For a rough count, we will sometimes consider multiplications arid divisions only, referring to those jointly as multiplicativeoperations. As an example, we examine the process ofinverting a $2\times 2$ matrix by elimination.

The whole process requires eight multiplicative operations. Note that we do not count operations with predictable results, such as 1a, 0a, a/a, 0/a.
a. How many multiplicative operations are required to invert a $3\times 3$ matrix by elimination1?
b. How many multiplicative operations are required to invert an $n\times n$ matrix by elimination?
c. If it takes a slow hand-held calculator 1 second to invert a $3\times 3$ matrix, how long will it take the same calculator to invert a $12\times 12$ matrix? Assume that the matrices are inverted by Gauss—Jordan elimination and that the duration of the computation is proportional lo the number of multiplications and divisions involved.