ZoomNotes for Linear Algebra

Part 5: Determinant of a Square Matrix

5.1 3 by 3 and $n$ by $n$ Determinants

The determinant is a number associated with every square matrix that determines whether the matrix is invertible ($\det A \neq 0$) or singular ($\det A = 0$).

3x3 Determinant (6 signed terms):

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

The three positive terms come from even permutations (identity, (1 2 3), (1 3 2)) and the three negative terms from odd permutations ((2 3), (1 2), (1 3)).

Three Defining Properties of Determinants:

1. Row exchange flips sign: If two rows are swapped, the determinant changes sign.
2. Linearity in each row:
   • Multiplying a row by a scalar multiplies the determinant by that scalar.
   • Adding two matrices that differ only in one row: determinant adds.
3. $\det I = 1$

Big Formula for $n \times n$:

$$\det A = \sum_{\text{permutations } P} (\det P) \, a_{1,\alpha_1} a_{2,\alpha_2} \cdots a_{n,\alpha_n}$$

There are $n!$ terms. Each term picks one entry from each row and each column, multiplies them, and assigns sign $+1$ for even permutations and $-1$ for odd permutations.

Immediate Consequences:

• If $A$ has a zero row, $\det A = 0$
• If two rows are identical, $\det A = 0$
• $\det$ of a triangular matrix = product of diagonal entries (pivots)
• $\det A = 0$ iff $A$ is singular

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5.2 Cofactors and the Formula for $A^{-1}$

Minor: $M_{ij}$ is the determinant of the $(n-1) \times (n-1)$ matrix obtained by removing row $i$ and column $j$.

Cofactor:

$$C_{ij} = (-1)^{i+j} M_{ij}$$

Cofactor Expansion (along row 1):

$$\det A = a_{11} C_{11} + a_{12} C_{12} + \cdots + a_{1n} C_{1n}$$

Expansion works along any row or column with the same result.

Inverse Formula via Cofactors:

$$A^{-1} = \frac{(\text{cofactor matrix})^T}{\det A}$$

The cofactor matrix (also called the adjugate or adjoint) has $C_{ij}$ in position $(i,j)$. Its transpose is the adjugate matrix.

For 2x2:

$$A^{-1} = \frac{1}{\det A} \begin{bmatrix} C_{11} & C_{21} \\ C_{12} & C_{22} \end{bmatrix} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

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5.3 $\det(AB) = (\det A)(\det B)$ and Cramer's Rule

The Product Rule:

$$\det(AB) = (\det A)(\det B)$$

This is the most important determinant property.

Corollaries:

• $\det A^{-1} = 1/\det A$
• $\det(A^k) = (\det A)^k$
• $\det(Q) = \pm 1$ for any orthogonal matrix $Q$
• $\det(U) = \prod$ of pivots for triangular $U$

Cramer's Rule: Solve $Ax = b$ using determinants:

$$x_j = \frac{\det B_j}{\det A}$$

where $B_j$ is $A$ with column $j$ replaced by $b$.

While theoretically elegant, Cramer's Rule is computationally expensive ($O(n!)$ operations via the big formula) and is never used for numerical computation in practice. Gaussian elimination ($O(n^3)$) is far more efficient.

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5.4 Volume of Box = $|\det E|$ where $E$ is Edge Matrix

Geometric Interpretation: The absolute value of the determinant equals the volume of the parallelepiped formed by the column (or row) vectors of the matrix.

For a 2x2 matrix $A = [\mathbf{a}_1 \; \mathbf{a}_2]$:

$$|\det A| = \text{area of parallelogram formed by } \mathbf{a}_1 \text{ and } \mathbf{a}_2$$

For a 3x3 matrix:

$$|\det A| = \text{volume of parallelepiped formed by } \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$$

Edge Matrix $E$: If we have vectors $\mathbf{e}_1, \dots, \mathbf{e}_n$ as the edges of a parallelepiped, then:

$$\text{Volume} = |\det(E)| \quad \text{where } E = [\mathbf{e}_1 \; \mathbf{e}_2 \; \cdots \; \mathbf{e}_n]$$

Why QR factorization makes this clear: $A = QR$ where $Q$ is orthogonal ($|\det Q| = 1$) and $R$ is triangular with pivots on the diagonal. The volume is $|\det R| = |r_{11} r_{22} \cdots r_{nn}|$, which is the product of base lengths and perpendicular heights.

Linear Transformation Interpretation: When a linear transformation $T$ represented by matrix $A$ acts on a region, volumes are multiplied by $|\det A|$:

$$\text{Vol}(T(\Omega)) = |\det A| \cdot \text{Vol}(\Omega)$$

This is the basis for the change-of-variables formula in multivariable calculus.