Recall that if A is an m x n matrix and B is a p x q matrix, then the product C $=$ AB is defined if and only if n $=$ p$,$ in which case C is an m x q matrix.

Each column of the matrix AB can be computed as the product of A times the corresponding column of B:

For instance, if A is $3$ x $4$ and B is $4$ x $5,$ the product is given by the matrix

i$.$e$.,$ where and are the jth columns of C and B respectively.

Write a function named columnproduct that takes as input two matrices A and B and a random number k$,$ and as output produces the product, C$,$ by columns of the two matrices and the intermediate value, z$,$ of the output matrix at the end of k iterations $($the matrix z is used by MATLAB Grader to determine whether your code is correct$).$

The function should work for any dimension of A and B and it should perform a check to see if the dimensions match and return an empty product matrix if the dimensions do not match.

Specific instructions for writing the function:

Extract the dimension of A using the command size and store the result in the variables m and n$.$

Extract the dimension of B using the command size and store the result in the variables p and q

Use an if statement to perform a check on the dimensions to determine whether the multiplication is defined

If the multiplication is defined:

initialize the vector C as a matrix of zeros of the appropriate dimension

use a single for loop to evaluate the product C $=$ AB$.$ The loop counter should be the variable j$.$ Each iteration of the for loop should evaluate the jth column of the matrix C as the product of A times the jth column of the matrix B$.$ Also, inside the for loop, after you compute the jth column of C$,$ insert the following commands:

if isequal$($j$,$k$)$

z $=$ C;

end

If the multiplication is not defined, display the message 'dimensions do not match' and return an empty C vector and an empty z vector.