Question

How to do matrix multiplication to find unknown matrix?

Answer 1

This is quite difficult to try to explain so I'm going to set up some matrices to demonstrate as best as I can.  If matrix A has 3 rows and 2 columns, it would be respresented as $A_{3*2}$.  This matrix can only be multiplied by another matrix that has the same number off rows as the number of columns in A.  For example, if matrix B has 2 rows and 3 columns, it would be represented as $B_{2*3}$. If we set them next to each other, it might be easier to see the rule: $A_{3*2} B_{2*3}$.  The 2's match, and the other numbers represent how your solution matrix will look.  Your solution matrix will be a 3x3.  Here is matrix A: $\left[\begin{array}{ccc}1&2\\3&4\\5&6\end{array}\right]$ and here's B: $\left[\begin{array}{ccc}1&2&3\\4&5&6\\\end{array}\right]$.  We can multiply these according to the rules.  The multiplication works like this: $\left[\begin{array}{ccc}1&2\\3&4\\5&6\end{array}\right] * \left[\begin{array}{ccc}1&2&3\\4&5&6\\\end{array}\right]$.  first row of A times first column of B: (1*1)+(2*4) = 9.  that goes into row 1 column 1 of your solution matrix.  Go va to row 1 in A but column 2 in B: (1*2)+(2*5) = 12.  That goes into row 1 column 2 of your solution matrix.  Next row 1 of A and column 3 of B: (1*3)+(2*6) = 15.  That goes into the first row column 3 of the solutiong matrix.  Now move to row 2 of A column 1 of B: (3*1)+(4*4) = 19.  That goes into row 2 column 1 in your solution matrix.  Next row 2 A, column 2 B: (3*2)+(4*5) = 26.  That goes into row 2 column 2 solution matrix. Continue as I showed you.  You should be fine.