Matrix multiplication looks strange the first time you meet it, because you do not multiply matching entries like you do when adding matrices. Instead, each entry of the answer combines a whole row from the first matrix with a whole column from the second. Once you see that pattern, the whole process becomes mechanical.
How it works
Two matrices can only be multiplied when the number of columns in the first equals the number of rows in the second. A 2 by 3 matrix times a 3 by 2 matrix works and gives a 2 by 2 answer. A 2 by 3 matrix times a 2 by 2 matrix does not work, because 3 does not equal 2. Each answer entry is the sum of the products of matching pairs: first entry of the row times first entry of the column, plus second times second, and so on.
Step by step
- Check the sizes: columns of the first matrix must equal rows of the second.
- Write down the size of the answer: rows of the first by columns of the second.
- Pick the top-left answer entry: take row 1 of the first matrix and column 1 of the second.
- Multiply matching pairs and add the results. That is your first entry.
- Move across the row, then down to the next row, repeating the same row-times-column step.
- Check your answer by testing one entry a second way, or by checking the size of the result.
Worked example
Multiply [[1, 2, 3], [4, 5, 6]] by [[7, 8], [9, 10], [11, 12]]. The sizes are 2 by 3 and 3 by 2, so the answer is 2 by 2. Top-left: 1×7 + 2×9 + 3×11 = 7 + 18 + 33 = 58. Top-right: 1×8 + 2×10 + 3×12 = 8 + 20 + 36 = 64. Bottom-left: 4×7 + 5×9 + 6×11 = 28 + 45 + 66 = 139. Bottom-right: 4×8 + 5×10 + 6×12 = 32 + 50 + 72 = 154. The answer is [[58, 64], [139, 154]].
Common mistakes
- Multiplying matching entries instead of rows by columns. That is addition, not multiplication.
- Trying to multiply matrices whose inner sizes do not match.
- Forgetting a term in the sum, especially with 3 by 3 matrices where each entry has three products.
- Mixing up rows and columns, which gives an answer of the wrong size.
Try it with the free tool
You can follow every step above by hand, but the free tool below does the same calculation instantly. Try your own numbers in it, then check one result by hand so you trust the method as well as the answer. Open the full tool here: open the tool.
Checklist before you finish
- Write down every input number first, with its unit, before you calculate anything.
- Work through the steps in order, and do not skip the unit conversions.
- Check the answer against the worked example above: is it the right size, and does the direction make sense?
- Round only at the end, and keep the unrounded figure for any follow-up calculation.
Practice with your own numbers
Pick three cases of your own and work each one by hand. First, try a simple round-number case you can check mentally. Second, try a realistic case with awkward numbers, the kind you will actually meet. Third, try an edge case, such as a very small or very large input, and confirm the answer still behaves sensibly. If any case gives a surprising result, re-check the units and the formula step by step before you trust it.
If your answer looks wrong
Most errors come from a small set of causes. Check that you used the same units throughout, that you divided where the formula divides and multiplied where it multiplies, and that you applied any percentage as a decimal. Check the order of operations: powers and brackets come before multiplication and division, which come before addition and subtraction. Finally, test the method on the worked example in this guide. If your method reproduces the example answer, the method is right and the inputs deserve a second look.
Quick tips
Cover all but one row and one column with your fingers or a ruler so you only see the pair you are working on. Work slowly on the first few examples; speed comes once the pattern is automatic.