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\(2 \times 2\) & \(3 \times 3\)
Determinants

Master the formulas and techniques for computing determinants of small matrices: the foundation of invertibility, area, and linear transformations.

Linear Algebra

What Is a Determinant?

A determinant is a single number computed from a square matrix. It tells you critical information about the matrix: whether it is invertible, how it scales area or volume, and the orientation of the transformation it represents.

Why Determinants Matter

Invertibility
A matrix \(A\) is invertible if and only if \(\det(A) \neq 0\).
Area & Volume
\(|\det(A)|\) gives the scaling factor for area (\(2 \times 2\)) or volume (\(3 \times 3\)).
Solving Systems
Cramer's Rule uses determinants to solve \(Ax = b\) when \(\det(A) \neq 0\).

Geometric Intuition

Unit Square
Area = 1
\(A\)
Transformed
\( \text{Area} = |\det(A)| \)
col 1
col 2

The \(2 \times 2\) Formula

\(a\)
\(b\)
\(c\)
\(d\)

Main vs. Anti-Diagonal

Main Diagonal
Runs top-left to bottom-right: entries \(a\) and \(d\). Their product \(ad\) gets the positive sign.
Anti-Diagonal
Runs top-right to bottom-left: entries \(b\) and \(c\). Their product \(bc\) gets the negative sign.

Worked Example: \(2 \times 2\)

Find the determinant of \(\begin{pmatrix} 3 & 7 \\ 1 & 5 \end{pmatrix}\).

1
Identify entries: \(a = 3\), \(b = 7\), \(c = 1\), \(d = 5\).
2
Main diagonal product: \(ad = 3 \times 5 = \color{#818cf8}{15}\).
3
Anti-diagonal product: \(bc = 7 \times 1 = \color{#fbbf24}{7}\).
4
Subtract: \(\det = \color{#818cf8}{15} - \color{#fbbf24}{7} = \color{#4ade80}{8}\).

The \(3 \times 3\) Determinant

For a \(3 \times 3\) matrix, we use cofactor expansion (also called expansion along a row or column). The most common approach expands along the first row.

Each first-row entry multiplies the determinant of its \(2 \times 2\) minor (the submatrix formed by deleting that entry's row and column), with alternating signs: \(+, -, +\).

Cofactor Expansion Visual

Sign Pattern Along Row 1
\(+a\)
times minor
\(e\)
\(f\)
\(h\)
\(i\)
\(ei - fh\)
\(-b\)
times minor
\(d\)
\(f\)
\(g\)
\(i\)
\(di - fg\)
\(+c\)
times minor
\(d\)
\(e\)
\(g\)
\(h\)
\(dh - eg\)

How to Compute Any Determinant

1
Check the size. For \(2 \times 2\), use the formula \(ad - bc\) directly.
2
For \(3 \times 3\): choose a row (usually the first) to expand along.
3
Apply the sign pattern \((+, -, +)\) to the entries of that row.
4
Find each \(2 \times 2\) minor by deleting the entry's row and column.
5
Multiply and sum: each signed entry times its minor's determinant.

Example: \(3 \times 3\) Determinant

Find \(\det\begin{pmatrix} 2 & 1 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 2 \end{pmatrix}\).

1
Expand along row 1: \(\color{#818cf8}{+2}\cdot M_{11} \;\color{#fbbf24}{-1}\cdot M_{12} \;\color{#f87171}{+3}\cdot M_{13}\).
2
\(M_{11} = \det\begin{pmatrix}4&5\\0&2\end{pmatrix} = 4 \cdot 2 - 5 \cdot 0 = \color{#818cf8}{8}\).
3
\(M_{12} = \det\begin{pmatrix}0&5\\1&2\end{pmatrix} = 0 \cdot 2 - 5 \cdot 1 = \color{#fbbf24}{-5}\).
4
\(M_{13} = \det\begin{pmatrix}0&4\\1&0\end{pmatrix} = 0 \cdot 0 - 4 \cdot 1 = \color{#f87171}{-4}\).
5
Combine: \(2(8) - 1(-5) + 3(-4) = 16 + 5 - 12 = \color{#4ade80}{9}\).

Example: Negative Entries

Compute \(\det\begin{pmatrix} -1 & 2 \\ 4 & -3 \end{pmatrix}\).

1
Identify: \(a = -1\), \(b = 2\), \(c = 4\), \(d = -3\).
2
Main diagonal: \(ad = (-1)(-3) = \color{#818cf8}{3}\).
3
Anti-diagonal: \(bc = (2)(4) = \color{#fbbf24}{8}\).
4
Result: \(\det = 3 - 8 = \color{#f87171}{-5}\).

Key Properties

Multiplicative
\(\det(AB) = \det(A) \cdot \det(B)\). The determinant of a product is the product of determinants.
Transpose
\(\det(A^T) = \det(A)\). Transposing doesn't change the determinant.
Scalar Factor
\(\det(kA) = k^n \det(A)\) for an \(n \times n\) matrix. Scaling every entry by \(k\) multiplies the det by \(k^n\).

Sarrus' Rule (Shortcut)

\(a\)
\(b\)
\(c\)
\(d\)
\(e\)
\(f\)
\(g\)
\(h\)
\(i\)
\(a\)
\(b\)
\(d\)
\(e\)
\(g\)
\(h\)
\(+ aei + bfg + cdh\)
\(- ceg - bdi - afh\)

Common Mistakes

✕ Wrong Subtraction Order
Writing \(bc - ad\) instead of \(ad - bc\). The main diagonal product always comes first.
✕ Forgetting the Sign Pattern
In \(3 \times 3\) cofactor expansion, the second term is subtracted, not added: \(+, -, +\).
✕ Wrong Minor Matrix
Deleting the wrong row or column when finding a \(2 \times 2\) minor. Always delete the entry's own row and column.

\(2 \times 2\) vs. \(3 \times 3\) Summary

\(2 \times 2\) \(3 \times 3\)
Formula \(ad - bc\) Cofactor expansion
Number of terms 2 6
Shortcut Direct formula Sarrus' Rule
Geometric meaning Signed area Signed volume
Singular when \(ad = bc\) \(\det = 0\)

Question 1 of 5

What is \(\det\begin{pmatrix} 4 & 6 \\ 3 & 2 \end{pmatrix}\)?

Question 2 of 5

If \(\det(A) = 0\), which statement is true?

Question 3 of 5

In cofactor expansion along the first row of a \(3 \times 3\) matrix, the sign pattern is:

Question 4 of 5

Compute \(\det\begin{pmatrix} 1 & 2 & 0 \\ 3 & 1 & 2 \\ 0 & 1 & 3 \end{pmatrix}\).

Question 5 of 5

If \(\det(A) = 5\) and \(\det(B) = -3\), what is \(\det(AB)\)?