How to Use the Rule of Sarrus to Find a 3×3 Determinant

What the rule of Sarrus is for

If you have been handed a 3×3 matrix and asked for its determinant, the rule of Sarrus is probably the friendliest way to get one. It turns the whole thing into six multiplications and one subtraction, laid out so you can read them off the page instead of holding them in your head.

It is named after Pierre Frédéric Sarrus, who published it in 1833. The reason it is still taught is that it is visual. Once the diagonals are drawn, the six pieces of the answer are sitting right there in front of you, and there is no table of plus and minus signs to keep track of.

There is one catch, and it is worth knowing up front: it only works on 3×3 matrices. Not 2×2, not 4×4. That is not a rule someone invented to be annoying — there is a real reason for it, and we come back to it at the end.

Step 1 — copy the first two columns

Write your matrix out. Then write the first column again to the right of it, and the second column after that. You now have a grid three rows tall and five columns wide.

Those two extra columns are not new information — they are the same numbers you already had. They are there so that every diagonal has three numbers to travel through. Without them, two of your six diagonals would run off the edge of the matrix after only two steps.

This is the step people skip, and it is the single most common reason a Sarrus calculation comes out wrong. If your answer only has four terms in it instead of six, this is almost certainly why.

Step 2 — multiply down the three diagonals, then add

Start at the first number in the top row. Move down and to the right, multiplying the three numbers you pass through. Then do the same starting from the second number in the top row, and again from the third.

That gives you three products. Add them together and set the total aside — you will need it in a moment.

These three are the positive part of the determinant. In the calculator on this page they are the coloured diagonals running down to the right.

Step 3 — multiply up the three diagonals, then add

Now do exactly the same thing from the bottom. Start at the first number in the bottom row, move up and to the right, and multiply the three numbers you pass through. Repeat from the second and third numbers of the bottom row.

Add those three products together too. This is your second total.

These are the terms that get subtracted. In the calculator they are the white diagonals running up to the right, so you can tell at a glance which group any term belongs to.

Step 4 — subtract

The determinant is your first total minus your second total. That is the whole method.

The order matters here, and it is easy to get backwards when you are working quickly: it is down-right total minus up-right total. If you subtract the other way round you will get the right number with the wrong sign, which is a frustrating way to lose marks on an otherwise correct answer.

How to tell whether you got it right

There is a quick sanity check that catches a lot of slips. If two rows of your matrix are identical, or if one row is just another row multiplied by some number, then the determinant has to be exactly zero. If you got anything else, one of your six products is wrong.

The other check is to work the same matrix through cofactor expansion and see whether you land on the same number. It takes another minute, but two methods agreeing is much stronger evidence than one method feeling right.

And if you would rather just confirm it: type your matrix into the calculator on this page. It shows all six products separately, so you can compare them against yours one at a time and find exactly where the two of you diverged.

The four mistakes worth watching for

• Forgetting the repeated columns. You end up with four terms instead of six, and no amount of careful arithmetic will save the answer.
• Subtracting in the wrong order. It is (down-right) − (up-right). Reversing it flips the sign of your final answer.
• Losing a minus sign inside a product. Two negatives in the same diagonal multiply to a positive. Writing each product on its own line, signs and all, costs a few seconds and prevents most of these.
• Reaching for it on a 4×4 matrix. It genuinely does not work there, and the answer it gives is not close — it is missing two thirds of the terms.