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Exercise 7 - Matrix Operation Count

Let Anxn be a given square matrix. Compute the number of multiplications and additions required to evaluate A28 using

a) the naive method, A28 = A x A x A ......... A (28 times)

b) A2, A4 = A2 x A2 etc.

Solution:

a) For the given square matrix,  A x A has

no of multiplication = n x n x n = n3

no of addition = (n-1) x n x n = n3 - n2

For total of 28 times,

no of multiplication = 27 n3

no of addition = 27(n3 - n2)

b) 

Step 1

A2 = A x A

Step 2

A4 = A2 x A2

Step 3

A8 = A4 x A4

Step 4

A16 = A8 x A8

Step 5

A12 = A8 x A4

Step 6

A28 = A16 x A12

Total no of steps required = 6

So, total no of multiplication and addition 6n3 and 6(n3 - n2) respectively.

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