# Video: Finding the Product of Two Matrices

Consider the matrices 𝐴 = −3, −4, 4 and −4, 4, 4 and 5, −1, −1 and 𝐵 = −2, −3, 2 and 6, 0, 2 and 3, 5, −4. Find 𝐴𝐵 if possible.

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### Video Transcript

Consider the matrices 𝐴 equals negative three, negative four, four, negative four, four, four, five, negative one, negative one and 𝐵 equals negative two, negative three, two, six, zero, two, and three, five, negative four. Find 𝐴𝐵 if possible.

We can multiply two matrices — if and only if — the number of columns in the first matrix equals the number of rows in the second. If this is not the case, the product does not exist. In this case, we say the product is undefined.

Since 𝐴 and 𝐵 are both three-by-three matrices, their product will also be a three-by-three matrix. To find the first entry in 𝐴𝐵, we find the sum of the products of the entries in row one of 𝐴 and column one of 𝐵. Negative three multiplied by negative two add negative four times six add four times three is negative six.

To find the entry for row one, column two of 𝐴𝐵, we again find the sum of the products of row one of 𝐴 and column two of 𝐵. Negative three multiplied by negative three add negative four multiplied by zero add four multiplied by five is 29.

To find the entry for row one, column three of 𝐴𝐵, we now find the sum of the products of row one of 𝐴 and column three of 𝐵. Negative three multiplied by two add negative four multiplied by two add four multiplied by negative four is negative 30.

Let’s repeat this process with row two of 𝐴 and column one of 𝐵. Adding together the products of the entries in row two of 𝐴 and column one of 𝐵 gives us a total of 44. The products of row two of 𝐴 and column two of 𝐵 give us a sum of 32. And finally, the products of row two of 𝐴 and column three of 𝐵 give us a total of negative 16.

To find the first entry in the final row of 𝐴𝐵, we find the sum of the products of row three of 𝐴 and column one of 𝐵. This gives us a value of negative 19. The entry in row three, column two of 𝐴𝐵 gives us a value of negative 20. And the sum of the products of row three of 𝐴 and column three of 𝐵 is 12.

The product of matrices 𝐴 and 𝐵 is, therefore, 𝐴𝐵 as shown.