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Determinant
Properties

Master the key properties of determinants: linearity, the effects of row operations, and how they connect to matrix invertibility.

Linear Algebra

What Is a Determinant?

The determinant is a scalar value assigned to every square matrix. For a \(2 \times 2\) matrix, it measures the signed area of the parallelogram formed by its column vectors.

x y
\(\mathbf{a} = (a_1, a_2)\)
\(\mathbf{b} = (b_1, b_2)\)
Area = \(|\det(A)|\)

Why Do Determinants Matter?

Invertibility
A matrix \(A\) is invertible if and only if \(\det(A) \neq 0\).
Volume Scaling
\(|\det(A)|\) gives the factor by which \(A\) scales areas (2D) or volumes (3D).
Eigenvalues
The characteristic polynomial \(\det(A - \lambda I) = 0\) finds eigenvalues.

Multilinearity in Each Row

The determinant is linear in each row when the other rows are held fixed. This means two things: it respects scalar multiplication and addition within a single row.

Three Row Operations

1
Swap two rows: Exchanging rows \(i\) and \(j\) negates the determinant: \(\det(A') = -\det(A)\).
2
Scale a row by \(c\): Multiplying row \(i\) by scalar \(c\) multiplies the determinant by \(c\): \(\det(A') = c \cdot \det(A)\).
3
Add a multiple of one row to another: Replacing \(\mathbf{r}_i\) with \(\mathbf{r}_i + k\mathbf{r}_j\) does not change the determinant.

The Multiplicative Property

This is one of the most powerful identities in linear algebra. It connects the determinant of a product to the product of determinants, and it follows from the multilinearity and alternating properties.

Corollary 1
\(\det(A^{-1}) = \dfrac{1}{\det(A)}\) when \(A\) is invertible.
Corollary 2
\(\det(A^n) = [\det(A)]^n\) for any positive integer \(n\).

Transpose and Determinant

This means every property that holds for rows also holds for columns. Swapping two columns negates the determinant, scaling a column scales it, etc.

A rows → cols Transpose
\(A^T\)
 det unchanged
\(\det(A) = \det(A^T)\)

Linearity vs. Multilinearity

A common source of confusion: the determinant is not a linear function of the matrix. It is multilinear: linear in each row separately.

Linear (FALSE) Multilinear (TRUE)
Sum \(\det(A+B) = \det(A)+\det(B)\) ✕ Row-wise additivity ✓
Scalar \(\det(cA) = c \cdot \det(A)\) ✕ \(\det(cA) = c^n \det(A)\) ✓
Scope Entire matrix at once One row at a time

Triangular Matrices

For any upper triangular, lower triangular, or diagonal matrix, the determinant is simply the product of the diagonal entries.

\(a_{11}\)
 ★ ★ 0
\(a_{22}\)
 ★ 0 0
\(a_{33}\)
 Upper Triangular →
\(\det(A) = a_{11} \cdot a_{22} \cdot a_{33}\)
 Product of diagonal entries

How to Compute Using Properties

1
Set up a tracker: Start with \(\det(A)\) and a multiplier \(m = 1\).
2
Eliminate below each pivot: Add multiples of one row to another. This keeps \(\det\) unchanged.
3
Track row swaps: Each swap flips the sign. Update \(m \leftarrow -m\).
4
Track row scaling: If you divide a row by \(c\), update \(m \leftarrow m \cdot c\).
5
Multiply diagonal: Once in triangular form, \(\det(A) = m \cdot \prod a_{ii}\).

Example 1: Row Swap on \(2 \times 2\)

Let \( A = \begin{pmatrix} 3 & 1 \\ 2 & 5 \end{pmatrix} \). Find \(\det(A)\) and then the determinant after swapping the two rows.

Example 2: Row Elimination on \(3 \times 3\)

Find the determinant of \( B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix} \).

Example 3: Scalar Multiplication

If \(\det(A) = 5\) for a \(3 \times 3\) matrix \(A\), what is \(\det(2A)\)?

Example 4: The Product Rule

Given \(\det(A) = 3\) and \(\det(B) = -2\), find \(\det(A^2 B^{-1})\).

Common Mistakes

✕ det is linear
Writing \(\det(A+B) = \det(A) + \det(B)\). This is false: the determinant is multilinear in rows, not linear in the whole matrix.
✕ Wrong scalar power
Writing \(\det(cA) = c \cdot \det(A)\) instead of \(c^n \det(A)\). Each of the \(n\) rows contributes a factor of \(c\).
✕ Forgetting swap sign
After a row swap, failing to negate the determinant. Each swap flips the sign: two swaps restore the original sign.

Property Summary

Property Formula Effect on det
Swap rows \(i, j\) \(\det(A') = -\det(A)\) Sign flip
Scale row by \(c\) \(\det(A') = c \cdot \det(A)\) Multiply by \(c\)
\(R_i + kR_j \to R_i\) \(\det(A') = \det(A)\) No change
Transpose \(\det(A^T) = \det(A)\) No change
Product \(\det(AB) = \det(A)\det(B)\) Multiplicative
Scalar matrix \(\det(cA) = c^n \det(A)\) Power of \(c\)

Question 1 of 5

If \(\det(A) = 7\) for a \(3 \times 3\) matrix, what is \(\det(3A)\)?

Question 2 of 5

Matrix \(B\) is obtained from \(A\) by swapping rows 1 and 3, then swapping rows 2 and 3. If \(\det(A) = 12\), what is \(\det(B)\)?

Question 3 of 5

Which row operation does not change the determinant?

Question 4 of 5

If \(\det(A) = -4\) and \(\det(B) = 3\), what is \(\det(A^{-1}B^2)\)?

Question 5 of 5

The determinant of an upper triangular \(4 \times 4\) matrix with diagonal entries \(2, -1, 3, 5\) is: