## Purpose of the

Row-down Method

The Row-down Method of Matrix Multiplication makes it a little easier to multiply the entries within a 'row' of a given matrix by the entries within a 'column' of a second matrix—what with a matrix 'row' being horizontal in nature and a matrix 'column' vertical.

Trying to multiply a horizontal matrix 'row' by a vertical matrix 'column' is like trying to play 3-dimensional chess! Hopefully, the illustrated steps below will turn the matrix-multiplication process into a game of checkers ...

## Illustrated Steps

### Two 2x2 square matrices A and B

We will be multiplying Matrix A by Matrix B.

As Matrix A has two columns and Matrix B two rows, our Product Matrix C is defined; and with each row of Matrix A and each column of Matrix B having the same number of 'entries,' our Product Matrix C will be structured with 2 rows and 2 columns.

### 2x2 square Product Matrix C

The top row of Product Matrix C will be using the first row from Matrix A for calculations, so add row 1 of Matrix A to each column of the first row of Product Matrix C.

The bottom row of Product Matrix C will be using the second row from Matrix A for calculations, so add row 2 of Matrix A to each column of the second row of Product Matrix C.

### 2x2 square Product Matrix C

To make it easier to multiply the rows you have just added to Product Matrix C by the columns from Matrix B (to be added): turn down the rows until they are vertical, looking like columns.

### 2x2 square Product Matrix C

Now that rows 1 and 2 of Matrix A have been positioned vertically on rows 1 and 2 of Product Matrix C, respectively, we are ready to add the columns from Matrix B.

### 2x2 square Product Matrix C

Add the first column from Matrix B to the first column of Product Matrix C, in positions C_{1,1} and C_{2,1} (to the right of turned-down Matrix A row one and Matrix A row two, respectively)—to act as each row's multiplier.

Next, add the second column from Matrix B to the second column of Product Matrix C, in positions C_{1,2} and C_{2,2} (to the right of turned-down Matrix A row one and Matrix A row two, respectively)—to act as each of those rows' multipliers.

### 2x2 square Product Matrix C

Finally, multiply each 'entry' listed within the turned-down Matrix A rows in each entry box of Product Matrix C by that row entrie's associated, horizontally-lined up, Matrix B column entry—and then add the sums of those entries' products together, vertically, within each Product Matrix C entry box ...

### 2x2 square Product Matrix C

... and *Voilà !*, as the French say; you have successfully calculated all of the entry values for Product Matrix C!