5. In 1975, J.B. Laderman discovered a non-commutative technique to multiply two 3 3 matrices with...

 

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5. In 1975, J.B. Laderman discovered a non-commutative technique to multiply two 3 3 matrices with 23 scalar products. (The conventional method employs 27.) Suppose Laderman's method is applied recursively to form the product of two general nxn matrices. a) Show how the n x n matrices would be partitioned into submatrices. How many submatrices; of what size? b) Give a recurrence for M(n), the number of scalar multiplications for the nxn case when Laderman's method is applied recursively using divide-and-conquer. Include an initial condition for M(1). c) Solve the recurrence in part b) to obtain a solution of the form M(n) = n. What is the exponent e? 5. In 1975, J.B. Laderman discovered a non-commutative technique to multiply two 3 3 matrices with 23 scalar products. (The conventional method employs 27.) Suppose Laderman's method is applied recursively to form the product of two general nxn matrices. a) Show how the n x n matrices would be partitioned into submatrices. How many submatrices; of what size? b) Give a recurrence for M(n), the number of scalar multiplications for the nxn case when Laderman's method is applied recursively using divide-and-conquer. Include an initial condition for M(1). c) Solve the recurrence in part b) to obtain a solution of the form M(n) = n. What is the exponent e?