6.1. Matrix Multiplication

Try these first

1. 1.

   What must be true about the dimensions of two matrices A and B for them to be multiplied?

   Hint

   Think about how many 'rows' one matrix can provide compared to the 'columns' of the other.

2. 2.

   What is the size of the resulting matrix when a 2x3 matrix is multiplied by a 3x4 matrix?

   Hint

   Consider the rows of the first matrix and columns of the second.

3. 3.

   Which of the following statements about matrix multiplication is true?

   • Matrix multiplication is commutative
   • Matrix multiplication is associative
   • Matrix multiplication is both commutative and associative

   Hint

   Think about whether you can swap the order of operations.

4. 4.

   If matrix A is 2x3 and matrix B is 3x4, what is the resulting matrix size of AB?

   Hint

   Remember the rule for matrix multiplication sizes.

5. 5.

   Given four matrices with dimensions 10x100, 100x5, 5x200, and 200x10, compute the minimum number of scalar multiplications needed to multiply them using a dynamic programming approach.

   Hint

   Analyze the costs of different groupings like (AB)(CD) compared to A(BCD).

6. 6.

   Propose and analyze a new recursive function for finding matrix multiplication costs and discuss its efficiency compared to the standard dynamic programming approach.

   Hint

   Think about how overlapping subproblems can reduce repeated calculations.