
so, the way a matrix multiplication works is
we multiply the LEFT-Matrix ROW by the RIGHT-Matrix COLUMN, while adding each value.
namely, the row of (4, -2, 3) times (-1, 3, 0) and the sum of each will give us the first value for the product matrix, and the multiplication goes like this -> (4)(-1)+(-2)(3)+(3)(0), that gives us a sum of -4-6+0 = -10, and that is the value that goes on the upper-left-corner of the product matrix.
so let's do the whole thing.
![\bf AB\implies \begin{bmatrix} 4&-2&3\\2&0&5 \end{bmatrix}\cdot \begin{bmatrix} -1&-4&3\\3&5&-3\\0&2&-2 \end{bmatrix} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lrlll} (4)(-1)+(-2)(3)+(3)(0)&=-10\\\\ (4)(-4)+(-2)(5)+(3)(2)&=-20\\\\ (4)(3)+(-2)(-3)+(3)(-2)&=12 \end{array}\impliedby \textit{the first row of the product} \\\\[-0.35em] \rule{34em}{0.25pt}](https://tex.z-dn.net/?f=%5Cbf%20AB%5Cimplies%20%5Cbegin%7Bbmatrix%7D%204%26-2%263%5C%5C2%260%265%20%5Cend%7Bbmatrix%7D%5Ccdot%20%5Cbegin%7Bbmatrix%7D%20-1%26-4%263%5C%5C3%265%26-3%5C%5C0%262%26-2%20%5Cend%7Bbmatrix%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cbegin%7Barray%7D%7Blrlll%7D%20%284%29%28-1%29%2B%28-2%29%283%29%2B%283%29%280%29%26%3D-10%5C%5C%5C%5C%20%284%29%28-4%29%2B%28-2%29%285%29%2B%283%29%282%29%26%3D-20%5C%5C%5C%5C%20%284%29%283%29%2B%28-2%29%28-3%29%2B%283%29%28-2%29%26%3D12%20%5Cend%7Barray%7D%5Cimpliedby%20%5Ctextit%7Bthe%20first%20row%20of%20the%20product%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D)
![\bf \begin{array}{lrll} (2)(-1)+(0)(3)+(5)(0)&=-2\\\\ (2)(-4)+(0)(5)+(5)(2)&=2\\\\ (2)(3)+(0)(-3)+(5)(-2)&=-4 \end{array}\impliedby \textit{the second row of the product} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \begin{bmatrix} -10&-20&12\\-2&2&-4 \end{bmatrix}~\hfill](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Barray%7D%7Blrll%7D%20%282%29%28-1%29%2B%280%29%283%29%2B%285%29%280%29%26%3D-2%5C%5C%5C%5C%20%282%29%28-4%29%2B%280%29%285%29%2B%285%29%282%29%26%3D2%5C%5C%5C%5C%20%282%29%283%29%2B%280%29%28-3%29%2B%285%29%28-2%29%26%3D-4%20%5Cend%7Barray%7D%5Cimpliedby%20%5Ctextit%7Bthe%20second%20row%20of%20the%20product%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D%5C%5C%5C%5C%20~%5Chfill%20%5Cbegin%7Bbmatrix%7D%20-10%26-20%2612%5C%5C-2%262%26-4%20%5Cend%7Bbmatrix%7D~%5Chfill)