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Determinants via
Row Reduction

Master how elementary row operations affect the determinant, and learn to compute \(\det(A)\) efficiently using row echelon form.

Linear Algebra

What is the Determinant?

The determinant is a single number assigned to every square matrix \(A\). It tells you whether the matrix is invertible, how it scales area/volume, and whether a system of equations has a unique solution.

\(\det(A) \neq 0\)
\(A\) is invertible; the system \(Ax = b\) has a unique solution.
\(\det(A) = 0\)
\(A\) is singular; the columns are linearly dependent.

Why Use Row Reduction?

Cofactor expansion on an \(n \times n\) matrix requires roughly \(n!\) operations. For a \(10 \times 10\) matrix that is over 3.6 million multiplications. Row reduction brings this down to about \(n^3\), making it the practical method for anything beyond \(3 \times 3\).

Cofactor Expansion
\(O(n!)\) operations: impractical for large \(n\).
Row Reduction
\(O(n^3)\) operations: fast and systematic.

Determinant of Triangular Matrices

If \(U\) is upper (or lower) triangular, then its determinant is the product of the diagonal entries:

\(u_{11}\)
\(u_{12}\)
\(u_{13}\)
\(0\)
\(u_{22}\)
\(u_{23}\)
\(0\)
\(0\)
\(u_{33}\)

The Three Row Operations

Each elementary row operation has a precise effect on the determinant. Memorize these three rules and you can compute any determinant by reduction.

Swap Two Rows
\(R_i \leftrightarrow R_j\): reverses the sign. Multiply \(\det\) by \(-1\).
Scale a Row
\(R_i \to k\,R_i\): multiplies \(\det\) by \(k\). Factor it out.
Add Multiple
\(R_i \to R_i + c\,R_j\): no change to \(\det\). Use freely!

The Core Formula

In practice, if you only use row replacement (\(R_i \to R_i + c\,R_j\)) and row swaps, the formula simplifies to:

Changes vs. No Change

The most critical distinction: which operations change the determinant and which leave it untouched.

Operation Effect on \(\det\) Must Track?
\(R_i \leftrightarrow R_j\) Multiply by \(-1\) Yes: count swaps
\(R_i \to k\,R_i\) Multiply by \(k\) Yes: track factor
\(R_i \to R_i + c\,R_j\) No change No

Step-by-Step Method

1
Write down \(A\). Initialize a swap counter \(s = 0\) and a scale factor accumulator \(k = 1\).
2
Forward elimination. Use \(R_i \to R_i + c\,R_j\) to create zeros below each pivot. Swap rows if the pivot is zero (increment \(s\)).
3
Reach upper triangular form \(U\). Read off the diagonal entries \(u_{11}, u_{22}, \ldots, u_{nn}\).
4
Compute: \(\det(A) = (-1)^s \cdot \frac{1}{k} \cdot u_{11}\,u_{22}\cdots u_{nn}\).

Example 1: \(2 \times 2\) Matrix

Find \(\det(A)\) where $$A = \begin{pmatrix} 3 & 1 \\ 6 & 4 \end{pmatrix}$$.

1
Eliminate below pivot: \(R_2 \to R_2 - 2R_1\) gives $$\begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$$. No swap, no scale.
2
Upper triangular! Diagonal: \(3\) and \(2\).
3
Result: \(\det(A) = (-1)^0 \cdot 3 \cdot 2 = \) \(6\).

Example 2: \(3 \times 3\) Matrix

Find \(\det(A)\) where $$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 7 \\ 3 & 7 & 11 \end{pmatrix}$$.

1
\(R_2 \to R_2 - 2R_1\), \(R_3 \to R_3 - 3R_1\): $$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & 1 & 2 \end{pmatrix}$$
2
\(R_3 \to R_3 - R_2\): $$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}$$
3
Diagonal: \(1, 1, 1\). Swaps: \(s = 0\).

Example 3: A Row Swap Is Needed

Find \(\det(A)\) where $$A = \begin{pmatrix} 0 & 1 & 2 \\ 3 & 4 & 5 \\ 6 & 7 & 9 \end{pmatrix}$$.

1
Pivot is 0! Swap \(R_1 \leftrightarrow R_2\): $$\begin{pmatrix} 3 & 4 & 5 \\ 0 & 1 & 2 \\ 6 & 7 & 9 \end{pmatrix}$$. Now \(s = 1\).
2
\(R_3 \to R_3 - 2R_1\): $$\begin{pmatrix} 3 & 4 & 5 \\ 0 & 1 & 2 \\ 0 & -1 & -1 \end{pmatrix}$$
3
\(R_3 \to R_3 + R_2\): $$\begin{pmatrix} 3 & 4 & 5 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$

Example 4: Scaling a Row

Find \(\det(A)\) where $$A = \begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}$$.

1
Optional: Scale \(R_1 \to \frac{1}{4}R_1\) for a cleaner pivot: $$\begin{pmatrix} 1 & \frac{1}{2} \\ 3 & 5 \end{pmatrix}$$. This multiplied \(\det\) by \(\frac{1}{4}\), so track \(k = \frac{1}{4}\).
2
\(R_2 \to R_2 - 3R_1\): $$\begin{pmatrix} 1 & \frac{1}{2} \\ 0 & \frac{7}{2} \end{pmatrix}$$
3
Diagonal: \(1 \cdot \frac{7}{2} = \frac{7}{2}\). Undo scaling: \(\det(A) = \frac{1}{1/4} \cdot \frac{7}{2} = 4 \cdot \frac{7}{2} = \) \(14\).

Visual Operation Summary

Matrix A

Swap Rows

\(\det \to -\det\)

Scale Row by \(k\)

\(\det \to k \cdot \det\)

Add \(c \cdot R_j\) to \(R_i\)

\(\det \to \det\) (unchanged)

U

triangular

Common Mistakes

✕ Forgetting Swap Signs
Each row swap flips the sign. Two swaps cancel out. Losing count gives the wrong sign.
✕ Ignoring Scale Factors
If you scale \(R_i \to kR_i\), the determinant was multiplied by \(k\). You must divide by \(k\) at the end.
✕ Thinking Row Addition Changes \(\det\)
\(R_i \to R_i + cR_j\) does NOT change the determinant. It is the "free" operation.

Summary Table

Step Action Effect on \(\det\)
1 Write matrix \(A\); set \(s=0\) None
2 Swap rows if \(pivot = 0\) \(\times (-1)\) each swap
3 \(R_i \to R_i + cR_j\) to eliminate None
4 Scale row (optional) \(\times k\)
5 Multiply diagonal of \(U\) \(\prod u_{ii}\)
6 Apply sign and scale correction \((-1)^s \cdot \frac{1}{k} \cdot \prod u_{ii}\)

Question 1 of 5

Which row operation does not change the value of the determinant?

Question 2 of 5

After row reduction (with no swaps and no scaling), a \(3 \times 3\) matrix becomes $$\begin{pmatrix} 2 & 5 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 7 \end{pmatrix}$$. What is \(\det(A)\)?

Question 3 of 5

During row reduction, you performed exactly 3 row swaps and no scaling. The resulting upper triangular matrix has diagonal entries \(1, 4, 2\). What is \(\det(A)\)?

Question 4 of 5

You scale row 1 by \(\frac{1}{5}\) (no swaps), then reduce to get $$U = \begin{pmatrix} 1 & 3 \\ 0 & 4 \end{pmatrix}$$. What is \(\det(A)\)?

Question 5 of 5

If \(\det(A) = 6\) and you swap two rows of \(A\) to get \(B\), then add \(3R_1\) to \(R_2\) to get \(C\), what is \(\det(C)\)?