We have seen that when given an m×n matrix A and an n×p matrix B, we can produce

Question

We have seen that when given an m×n matrix A and an n×p matrix B, we can produce AB using dot products (i.e., inner products). If we think of A has consisting of n column vectors and B as consisting of n row vectors, we can produce AB using the outer product. That is, if

then

Pay attention to what this says. This is the product of the first column of A with the first row of B plus the product of the second column of A with the second row of B and so forth. Let

Produce AB using

(a) normal matrix multiplication

(b) the outer product.

In both cases, show enough work to make it possible to see how the two approaches work and why they must produce the same result.

Explanation / Answer

a) AB by using normal matrix multiplication

by taking first matrix row & second matrix column ,first matrix first row to second matrix first column become first element of new matrix,similarly first matrix first row multiplying with second matrix second column become the second elemnet of first row of new matrix

AB= ae+bh af+bi ag+bj

ce+dh cf+di cg+dj

b) the outer product rule is by performing the dot product of first matrix column vector & second matrix row vector

column vectors are (a,c) and (b,d)

row vectors are (e,f,g) and (h,i,j)

ae+bh af+bi ag+bj

ce+dh cf+di cg+dj

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