Sarrus' Rule

Sarrus' Rule is a shortcut for finding the determinant of a 3x3 matrix. In Linear Algebra, you use it to multiply diagonals in a visual pattern instead of expanding by minors.

What is Sarrus' Rule?

Sarrus' Rule is a quick way to compute the determinant of a 3x3 matrix in Linear Algebra. If you have a matrix with three rows and three columns, this rule gives you a visual pattern for combining products of entries so you can get the determinant without doing a full cofactor expansion.

The setup is simple. Write the 3x3 matrix, then copy the first two columns to the right of it. That extra space lets you trace three diagonals that slope down and to the right, and three diagonals that slope up and to the right. Multiply the entries on each diagonal, add the three downward products, and subtract the three upward products.

For a matrix

$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

Sarrus' Rule gives

$$\det(A)=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-(a_{13}a_{22}a_{31}+a_{12}a_{21}a_{33}+a_{11}a_{23}a_{32}).$$

That formula is the same thing the visual method is doing, just written out in symbols.

A compact example makes the pattern easier to see. For

$$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$

Sarrus' Rule gives downward products of 1\cdot5\cdot9, 2\cdot6\cdot7, and 3\cdot4\cdot8, then subtracts the upward products 3\cdot5\cdot7, 2\cdot4\cdot9, and 1\cdot6\cdot8. The result is 0, which matches the fact that the rows are linearly dependent.

The biggest limitation is that Sarrus' Rule only works for 3x3 matrices. You cannot extend it to 4x4 matrices by copying more columns and tracing more diagonals. For larger matrices, you switch to cofactor expansion, row reduction, or another determinant method.

A good way to think about the rule is that it is a memory aid for one special determinant size. It is fast, visual, and easy to check, which is why it shows up so often in early matrix work.

Why Sarrus' Rule matters in Linear Algebra and Differential Equations

Sarrus' Rule matters because determinants show up all over Linear Algebra, and 3x3 matrices are one of the first places where the arithmetic gets big enough that a shortcut saves time. If you are checking whether a matrix is singular, solving a system, or working with geometry questions about area and volume scaling, you often need the determinant of a 3x3 matrix fast.

It also gives you a concrete picture of what a determinant calculation is doing. Instead of treating the determinant like a mysterious formula, you see it as a structured sum of products with a subtraction step that balances the orientation of the matrix entries. That visual pattern helps many students remember the order of terms and the sign pattern.

Sarrus' Rule is useful before you move on to cofactor expansion and row reduction because it builds confidence with 3x3 determinants. Once you can do this cleanly, you are better prepared for bigger determinant methods, where the calculations are longer but the same basic ideas about size, sign, and dependence still matter.

It also connects directly to later topics like singular matrices and linear dependence. If the determinant comes out 0, that is a signal that the rows or columns are not independent, and that changes whether the matrix can be inverted.

How Sarrus' Rule connects across the course

Determinant

Sarrus' Rule is just one method for finding a determinant, and it only applies when the matrix is 3x3. The determinant is the value you are actually computing, so if you do not know what the determinant means, Sarrus' Rule can feel like a memorized trick. This rule is one way to get that value quickly and check whether your matrix is singular or not.

Matrix

You use Sarrus' Rule on a matrix, not on a list of numbers or a vector. The rule depends on the matrix having exactly three rows and three columns, because the diagonal pattern only works in that shape. If the matrix is not square, or if it is larger than 3x3, you need a different determinant method.

Cofactor Expansion

Cofactor expansion is the more general determinant method, while Sarrus' Rule is the faster 3x3 shortcut. Both give the same determinant, but cofactor expansion works for any square matrix. If you know Sarrus' Rule first, cofactor expansion feels less random because you already understand the sign pattern and the idea of combining products with subtraction.

singular matrix

A singular matrix has determinant 0, and Sarrus' Rule is one of the quickest ways to check that for a 3x3 matrix. If your result is 0, the matrix is singular, which means it cannot be inverted. That connection is why determinant practice often leads right into singular versus invertible matrices.

Is Sarrus' Rule on the Linear Algebra and Differential Equations exam?

A quiz or problem set question will usually give you a 3x3 matrix and ask for its determinant, an inverse check, or a singular versus nonsingular decision. Sarrus' Rule is the move you use when the matrix is exactly 3x3 and you want a fast, reliable calculation method. Write the first two columns again, multiply the three downward diagonals, subtract the three upward diagonals, and simplify carefully. The most common mistake is mixing up the sign pattern or accidentally trying to use the rule on a 4x4 matrix. If a problem asks whether a matrix is singular, your Sarrus' Rule answer of 0 is the signal you need.

Sarrus' Rule vs Cofactor Expansion

These are easy to mix up because both methods compute determinants. Sarrus' Rule is only for 3x3 matrices and uses a diagonal pattern, while cofactor expansion works for any square matrix and is based on minors and alternating signs. If the matrix is bigger than 3x3, Sarrus' Rule is not an option.

Key things to remember about Sarrus' Rule

• Sarrus' Rule is a shortcut for finding the determinant of a 3x3 matrix.

• You extend the matrix by copying the first two columns to the right, then trace diagonal products.

• Add the three downward diagonal products and subtract the three upward diagonal products.

• The rule only works for 3x3 matrices, so it cannot be used on 2x2 or 4x4 matrices.

• If the determinant is 0, the matrix is singular, which often shows up in later matrix questions.

Frequently asked questions about Sarrus' Rule

What is Sarrus' Rule in Linear Algebra and Differential Equations?

Sarrus' Rule is a fast method for finding the determinant of a 3x3 matrix. You copy the first two columns to the right, multiply the diagonals going down, and subtract the diagonals going up. It is a calculation shortcut, not a general determinant method.

Does Sarrus' Rule work for 4x4 matrices?

No. Sarrus' Rule only works for 3x3 matrices. A common mistake is trying to stretch the visual pattern to larger matrices, but that does not give a correct determinant. For 4x4 and larger matrices, you use cofactor expansion or row reduction.

How do you use Sarrus' Rule step by step?

Write the 3x3 matrix, then repeat the first two columns beside it. Find the three products on the diagonals that slope down to the right, then find the three products on the diagonals that slope up to the right. Add the first group and subtract the second group.

Why does Sarrus' Rule matter when checking for a singular matrix?

A matrix is singular when its determinant is 0, and Sarrus' Rule gives you a quick way to test that for a 3x3 matrix. If your result is zero, the matrix cannot be inverted. That makes the rule useful in later matrix problems, not just determinant drills.