Properties of 3x3 Determinants

The good news about the $3 \times 3$ determinant: every property you learned for $2 \times 2$ carries over unchanged. The product rule, the transpose rule, row swaps, row scalings, row additions, they all hold, with the same proofs, the same applications, the same picture. This section lists them in one place and shows how to use them to shortcut what would otherwise be a six-term expansion.

The seven properties

1. Multiplicativity: $\det(AB) = \det(A) \det(B)$.
2. Transpose: $\det(A^T) = \det(A)$.
3. Row swap: swapping two rows (or two columns) multiplies $\det$ by $-1$.
4. Row scaling: multiplying one row by $c$ multiplies $\det$ by $c$.
5. Row addition: adding a multiple of one row to another leaves $\det$ unchanged.
6. Dependence: if two rows (or columns) are equal or proportional, $\det = 0$.
7. Scalar matrix: $\det(cA) = c^3 \det(A)$ for any $3 \times 3$ matrix.

The last property gives away the pattern. For an $n \times n$ matrix, $\det(cA) = c^n \det(A)$. One factor of $c$ per row. The exponent is the size of the matrix.

• Numerical Linear Algebra: Detecting whether a 3x3 matrix is invertible is a single determinant evaluation; LAPACK (the foundation of NumPy, MATLAB, R) uses row operations to reduce the matrix to triangular form, then multiplies diagonals.
• Quantum Chemistry: Wave-function determinants (Slater determinants) describe multi-electron systems; the antisymmetry property follows from the determinant's sign-flip under row swaps.
• Engineering Mechanics: Stiffness matrices of finite-element models must have non-zero determinant for the system to be stable; engineers check for ill-conditioning by monitoring the determinant during assembly.

(Use the widget to verify property 6 by hand: set two columns equal, watch the parallelepiped collapse to a parallelogram, det drops to $0$. Property 4: double the entries of any one column and the volume doubles. Property 7: double all three columns and the volume is $2^3 = 8$ times larger.)

Triangular shortcuts

For a triangular matrix (upper or lower), the determinant is just the product of the diagonal entries:

\det\begin{pmatrix} a_{11} & * & * \\ 0 & a_{22} & * \\ 0 & 0 & a_{33} \end{pmatrix} = a_{11} a_{22} a_{33}.

This is the fastest possible $3 \times 3$ determinant calculation. Triangular matrices show up after Gaussian elimination, so reducing first and then reading off the diagonal is faster than expanding cofactors on the original matrix.

Row reduction strategy

To compute any $3 \times 3$ determinant efficiently:

1. Use row addition (property 5) to clear entries below the diagonal, no sign change, no scaling factor.
2. If you need to swap rows to get a non-zero pivot, track each swap with a $-1$ factor.
3. If you scale a row by $c$, divide your eventual answer by $c$ (or carry the $c$ factor explicitly).
4. Read off the diagonal product, apply the accumulated sign / scale corrections.

This is how the algorithm in your widget actually computes the determinant under the hood. It is also how a computer evaluates a $100 \times 100$ determinant: row operations to triangular form, then a diagonal product.

Smart cofactor expansion

If a row or column already has zeros, exploit them. The cofactor expansion can be done along any row or column, pick the one with the most zeros, and entire terms in the expansion vanish before you compute anything. A row with two zeros leaves just one non-trivial cofactor; one with three zeros leaves zero. Choosing well can turn a six-term expansion into a one-term expansion.

Try it

• If $\det(A) = 4$ for a $3 \times 3$ matrix, compute $\det(2A)$. (Answer: $2^3 \cdot 4 = 32$.)
• Compute \det\begin{pmatrix} 3 & 0 & 0 \\ 5 & 2 & 0 \\ 1 & 4 & 7 \end{pmatrix} in one step.
• Suppose row 3 of a $3 \times 3$ matrix equals row 1 plus twice row 2. What is the determinant, and why?
• You swap rows 1 and 3, then multiply row 2 by $4$. If the original determinant was $10$, what is the new one? (Two operations: sign flip, then $\times 4$.)

Pause: property 2 says $\det(A^T) = \det(A)$. So every row property has a column counterpart. Why is the transpose rule so important for the rest of the chapter?

A trap to watch for

Property 7, $\det(cA) = c^n \det(A)$, surprises beginners every time. Students see $\det(2A) = 2 \det(A)$ for $2 \times 2$ and assume the same for $3 \times 3$. Wrong. For a $3 \times 3$ matrix, $\det(2A) = 2^3 \det(A) = 8 \det(A)$. The exponent is the size of the matrix, because scaling the matrix by $c$ means scaling every row by $c$, and each row contributes its own factor.

A second trap: forgetting to track sign changes during row reduction. If you swap rows twice during a calculation, you have $(-1)^2 = +1$, no net change. If you swap three times, you have $(-1)^3 = -1$. Beginners often track the first swap and forget the rest, getting a sign error in the final answer. The cleanest practice: write down every swap as you make it.

What you now know

You can apply seven properties of the $3 \times 3$ determinant, shortcut the calculation via row reduction or strategic cofactor expansion, and recognise zero determinants from linear dependence at sight. The next section is the payoff for all this determinant arithmetic: Cramer's rule, an explicit formula for solving a linear system in terms of determinants.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 17, §5, the properties listed and proved for the $n = 3$ case.
• Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. §5.1, row operations on determinants and the $c^n$ scaling rule.
• Hoffman, K.; Kunze, R. (1971). Linear Algebra (2nd ed.). Prentice-Hall. Chapter 5: the alternating multilinear characterisation that makes all seven properties one fact.
• Axler, S. (2015). Linear Algebra Done Right (3rd ed.). Springer. Chapter 10: product rule and dependence in operator language.