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Theproductoftwon times nmatricesX andY isX times Y = XY , whereXY is also an n times n matrix. ( XY ) ij

Theproductoftwon\times nmatricesX andY isX\times Y =XY,whereXY is also an n \times n matrix. (XY )ij (that is, the element at index i, j) can be computed as the dot product of the i-th row of X and the j-th column of Y .
The standard (brute-force) algorithm for matrix multiplication performs \Theta (n3) operations: each of the n2 values in XY must be computed, and each dot product takes \Theta (n) time to compute. Can we do better by using Divide and Conquer, la Karatsuba?
Matrix multiplication lends itself nicely to divide and conquer, because we can perform it blockwise. That is, we can carve X into 4( n2)\times ( n2) blocks as follows:
[]
X=AB CD
where A represents all values in the top left quadrant of the matrix, B represents all values in the top right quadrant, etc.
Lets carve Y into similar blocks: []
Y=EF GH
Now, we can express their product XY as follows: [][][]
XY= A B E F = AE+BG AF+BH C D G H CE+DG CF+DH
Also, as a reminder, adding two n \times n matrices X + Y takes \Theta (n2) time because for each pair of indices i,j,(X + Y)ij = Xij + Yij. Subtractin

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