 Here's a quick overview of matrix multiplication. A matrix is a series of numbers arranged in rows and columns. This matrix has two rows and three columns. It's a 2 by 3 matrix. This matrix has three rows and two columns. It's a 3 by 2 matrix. In order to multiply the matrices, they must be compatible. That means that the number of columns in the first matrix must be the same as the number of rows in the second matrix. The results of the multiplication will have the same number of rows as the first matrix and the same number of columns as the second matrix. In this example, the result matrix will be 2 by 2. The entry in row 1 column 1 of the result matrix will be the dot product of row 1 and column 1 in the first and second matrices. Dot product is a fancy name for some of the products of the corresponding entries in a row and column. That means that row 1 column 1 of our result will be 1 times 7 plus 2 times 8 plus 3 times 9. That adds up to 50. Row 1 column 2 of the result is the dot product of row 1 and column 2. That's 1 times 10 plus 2 times 11 plus 3 times 12. 10 plus 22 plus 36 adds up to 68. Row 2 column 1 is the dot product of row 2 and column 1. 4 times 7 plus 5 times 8 plus 6 times 9. 28 plus 40 plus 54 adds up to 122. Our last entry in the result row 2 column 2 is the dot product of row 2 and column 2 in the first and second matrices. That's 4 times 10 plus 5 times 11 plus 6 times 12. 40 plus 55 plus 72 adds up to 167. And that's the result of multiplying this matrix times that matrix. To do this in Java, you'll need a triple nested loop. The outer loop will iterate through the rows of the first matrix. The middle loop will go through the columns for the second matrix. And the inner loop will produce the dot product of the row and column you're currently looking at.