The Rule of Sarrus: The Fastest Method for 3×3 Matrices

The determinant of a 3×3 matrix is a single number you calculate from the nine entries arranged in three rows and three columns. For a 3×3 matrix, the Rule of Sarrus is the quickest method—it takes about two minutes once you know the pattern, and you only need basic multiplication and subtraction.

The Rule of Sarrus works by writing your matrix, then repeating the first two columns to the right of it. You multiply along three diagonals going down-right, add those products, subtract the products from three diagonals going down-left, and you have your answer. This method is faster than the cofactor expansion taught in many textbooks because it avoids nested calculations.

Key Takeaways

  • Write your 3×3 matrix, then repeat the first two columns to the right to create a 3×5 grid.
  • Multiply the three diagonals that slope downward from left to right, then add those three products together.
  • Multiply the three diagonals that slope downward from right to left, then add those three products together.
  • Subtract the second sum from the first sum to get your determinant.
  • A determinant of zero means the matrix cannot be inverted; any other number means it can be.

Setting Up Your Matrix for the Rule of Sarrus

Start with your 3×3 matrix written in standard form. For this example, use:

2 3 1 4 1 5 3 2 1

Write the matrix, then when ready to its right, write the first column again, then the second column again. Your extended grid now looks like this:

2 3 1 | 2 3 4 1 5 | 4 1 3 2 1 | 3 2

The vertical line is just a visual marker—it is not part of the calculation. You now have five columns and three rows. This layout is the entire setup; everything else is multiplication and addition.

Multiply the Down-Right Diagonals and Add Them

A down-right diagonal starts at the top and moves down and to the right. In your extended grid, there are exactly three of them. Trace each one and multiply the three numbers along it.

First diagonal: Start at the top-left (2), move down-right to the middle (1), then down-right to the bottom-right (1). Multiply: 2 × 1 × 1 = 2.

Second diagonal: Start at position (row 1, column 2), which is 3. Move down-right to 5, then down-right to 3. Multiply: 3 × 5 × 3 = 45.

Third diagonal: Start at position (row 1, column 3), which is 1. Move down-right to 4, then down-right to 2. Multiply: 1 × 4 × 2 = 8.

Add these three products: 2 + 45 + 8 = 55. Write this number down—you will use it in the final step.

Multiply the Down-Left Diagonals and Add Them

A down-left diagonal starts at the top and moves down and to the left. There are also exactly three of these. Trace each one and multiply the three numbers along it.

First diagonal: Start at position (row 1, column 3), which is 1. Move down-left to 1, then down-left to 3. Multiply: 1 × 1 × 3 = 3.

Second diagonal: Start at position (row 1, column 4)—remember, this is the repeated first column, so the value is 2. Move down-left to 5, then down-left to 2. Multiply: 2 × 5 × 2 = 20.

Third diagonal: Start at position (row 1, column 5), which is 3. Move down-left to 4, then down-left to 1. Multiply: 3 × 4 × 1 = 12.

Add these three products: 3 + 20 + 12 = 35. Write this number down as well.

Subtract to Get Your Determinant

Subtract the down-left sum from the down-right sum. In this example: 55 − 35 = 20.

The determinant of your matrix is 20. That is your final answer. If you had arrived at zero, the matrix would be singular (non-invertible). Any other number, positive or negative, means the matrix has an inverse and can be used in further calculations.

Common Mistakes to Avoid

The most frequent error is forgetting to repeat the first two columns. Without them, you cannot trace the diagonals correctly and your answer will be wrong. Always write out the extended 3×5 grid before you start multiplying.

Another common mistake is mixing up the direction of the diagonals. Down-right means you move down one row and right one column with each step. Down-left means you move down one row and left one column. If you trace a diagonal and end up outside the grid, you have chosen the wrong direction.

A third error is arithmetic—especially when one of your matrix entries is negative. Write out each multiplication step separately and double-check your sums before subtracting. A single wrong digit in one product will throw off your final answer.

When to Use the Rule of Sarrus Versus Other Methods

The Rule of Sarrus is the fastest method for hand calculation of 3×3 determinants and is the one taught in most linear algebra courses for this reason. It requires no memorization of cofactor patterns and works the same way every time.

If you are working with a 2×2 matrix instead, use the simpler formula: multiply the top-left and bottom-right entries, then subtract the product of the top-right and bottom-left entries. For matrices larger than 3×3, the Rule of Sarrus does not work—you will need cofactor expansion or a calculator. For 3×3 matrices, however, Sarrus is your best choice.

Frequently Asked Questions

What does a determinant of zero mean?

A determinant of zero means the matrix is singular and cannot be inverted. In practical terms, if you are using the matrix to solve a system of equations, a zero determinant means the system has either no solution or infinitely many solutions, not a unique answer.

Can the determinant be negative?

Yes. A negative determinant is perfectly valid and straightforward means the matrix reverses the orientation of space (in geometric terms, it reflects). The sign does not indicate an error—it is part of the correct answer.

Do I have to use the Rule of Sarrus, or can I use cofactor expansion instead?

You can use either method and arrive at the same answer. Cofactor expansion involves picking a row or column, multiplying each entry by its cofactor, and adding the results. It works but requires more steps and is easier to make arithmetic mistakes on. Sarrus is faster for 3×3 matrices.

What if my matrix has fractions or decimals in it?

The Rule of Sarrus works exactly the same way. Multiply the entries along each diagonal just as you would with whole numbers. Keep fractions in fraction form until the very end to avoid rounding errors, or use a calculator if decimals are involved.

How do I know if I calculated the determinant correctly?

Check your work by recalculating the down-right and down-left sums separately, then subtract again. If you get the same answer twice, it is correct. You can also verify using an online matrix calculator, but working through it by hand twice is the most reliable way to catch arithmetic errors.