Determinants, Inverses, and Volume

The determinant \(\det A\) is a scalar measuring oriented volume scaling under the linear map \(A\). It decides invertibility and appears in change-of-variables formulas, multivariate Gaussians, and geometric algorithms.

Diagram: area scaling

  unit square              after A (2D)

  ┌───┐                    ╱╲
  │   │   ──A──►          ╱  ╲
  └───┘                  ╱____╲
  area 1                 area |det A|

1. Invertibility criteria

For square \(A\in\mathbb{R}^{n\times n}\), the following are equivalent (TFAE):

1. \(A\) is invertible (exists \(B\) with \(AB=BA=I\))
   
2. \(\det A\neq 0\)
   
3. Columns form a basis of \(\mathbb{R}^n\) (linearly independent)
   
4. Rows form a basis of \(\mathbb{R}^n\)
   
5. \(\mathrm{rank}(A)=n\)
   
6. \(\ker(A)=\{0\}\)
   
7. \(0\) is not an eigenvalue of \(A\)
   
8. \(Ax=b\) has a unique solution for every \(b\)

Inverse: \(A^{-1}A=I=AA^{-1}\) when invertible.

Worked example 1 — \(2\times 2\) inverse

\(A=\begin{bmatrix}2&1\\0&3\end{bmatrix}\), \(\det A=6\neq 0\).

\[ A^{-1}=\frac{1}{6}\begin{bmatrix}3&-1\\0&2\end{bmatrix} =\begin{bmatrix}1/2&-1/6\\0&1/3\end{bmatrix}. \]

Verify \(AA^{-1}=I\).

Worked example 2 — singular

\(A=\begin{bmatrix}1&2\\2&4\end{bmatrix}\), \(\det=0\), columns collinear, \(\ker\) spanned by \((2,-1)\), not invertible.

2. Computing determinants

\(2\times 2\)

\[ \det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc. \]

\(3\times 3\) (cofactor expansion along row 1)

\[ \det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} =a(ei-fh)-b(di-fg)+c(dh-eg). \]

Via row reduction

Row-reduce \(A\) to upper triangular \(U\) with pivots \(u_{11},\ldots,u_{nn}\):

\[ \det A = (-1)^s \Bigl(\prod_i u_{ii}\Bigr) \Big/ \text{(scale factors)}, \]

more carefully tracking:

Row operation	Effect on det
Swap two rows	multiply by \(-1\)
Scale a row by \(k\)	multiply by \(k\)
Add multiple of one row to another	unchanged

Product of pivots (with sign from swaps) gives \(\det\) if you only used swaps and shears (no row scaling), or adjust for scalings.

Multiplicativity and friends

\[ \det(AB)=\det(A)\det(B),\qquad \det(A^\top)=\det(A),\qquad \det(A^{-1})=\frac{1}{\det A}\ (A\text{ invertible}), \]

\[ \det(cA)=c^n\det A,\qquad \det(A^{-1}BA)=\det B \]

(similarity preserves det — eigenvalues product invariant).

Worked example 3 — triangular

\[ \det\begin{bmatrix}1&2&0\\0&3&4\\0&0&5\end{bmatrix}=1\cdot 3\cdot 5=15. \]

Worked example 4 — product

If \(\det A=2\), \(\det B=-3\), then

\[ \det(A^2 B^{-1})=\det(A)^2\det(B)^{-1}=4\cdot\Bigl(-\frac13\Bigr)=-\frac43. \]

3. Geometric meaning

• \(|\det A|=\) volume of the parallelepiped spanned by the columns of \(A\)
  
• \(\det A>0\): orientation-preserving; \(\det A<0\): orientation-reversing (includes a reflection component)
  
• \(\det A=0\): columns collapse volume to a lower-dimensional flat
  
• Orthogonal matrices: \(Q^\top Q=I\) ⇒ \(\det Q=\pm 1\) (rotations vs improper rotations)

Worked example 5 — shear preserves area

\(S=\begin{bmatrix}1&k\\0&1\end{bmatrix}\), \(\det S=1\). Shears preserve area; shape changes.

Worked example 6 — scaling

\(D=\mathrm{diag}(\lambda_1,\ldots,\lambda_n)\), \(\det D=\prod\lambda_i\) — product of axis scale factors.

4. Determinant and eigenvalues

Characteristic polynomial \(p_A(\lambda)=\det(A-\lambda I)\) (sign convention variants exist).

\[ \det A = \prod_{i=1}^n \lambda_i \]

(over \(\mathbb{C}\), counting algebraic multiplicity). Trace \(\mathrm{tr}(A)=\sum\lambda_i\).

So \(\det A\neq 0\) iff no zero eigenvalue — matches invertibility.

5. Cramer’s rule (theory / tiny \(n\))

For invertible \(Ax=b\),

\[ x_i=\frac{\det(A_i)}{\det A}, \]

where \(A_i\) is \(A\) with column \(i\) replaced by \(b\).

Practice: use elimination / library solvers for \(n\gtrsim 3\). Cramer is exponential cost if determinants are expanded naively and numerically unstable if implemented poorly.

6. Explicit inverse formulas

\(2\times 2\)

\[ \begin{bmatrix}a&b\\c&d\end{bmatrix}^{-1} =\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}. \]

Adjugate (general \(n\))

\(A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)\) where \(\mathrm{adj}\) is the cofactor matrix transpose. Useful in proofs; rarely the right computational tool.

7. Jacobians and change of variables

If \(T:\mathbb{R}^n\to\mathbb{R}^n\) is differentiable and \(y=T(x)\), volumes transform by \(|\det DT(x)|\):

\[ \int_{T(U)} f(y)\,dy = \int_U f(T(x))\,|\det DT(x)|\,dx. \]

In probability: if \(Y=g(X)\) invertible, densities pick up \(|\det Dg^{-1}|\).

Worked example 7 — polar (idea)

\(x=r\cos\theta\), \(y=r\sin\theta\), Jacobian determinant magnitude is \(r\) — area element \(r\,dr\,d\theta\).

8. Multivariate Gaussians

For \(X\sim\mathcal{N}(\mu,\Sigma)\) with \(\Sigma\succ 0\),

\[ p(x)=(2\pi)^{-n/2}(\det\Sigma)^{-1/2} \exp\Bigl(-\tfrac12(x-\mu)^\top\Sigma^{-1}(x-\mu)\Bigr). \]

\(\det\Sigma\) scales the normalizing constant; ill-conditioned \(\Sigma\) (tiny eigenvalues) makes density peaks sharp and numerics fragile.

9. Numerical wisdom

1. Do not compute \(x=A^{-1}b\) by forming \(A^{-1}\) explicitly when you only need \(x\) — solve \(Ax=b\)
   
2. Determinants of large matrices can overflow/underflow; work in log-domain (\(\sum\log|\mathrm{pivots}|\))
   
3. Near-singular matrices: \(\det\approx 0\) is a symptom; prefer condition number \(\kappa(A)\) and residual checks
   
4. Integer matrices can have huge dets (Hadamard bound); exact arithmetic may need big integers

10. CS / graphics / systems connections

Domain	Role of det / inverse
Computer graphics	orientation tests, back-face, volume of tetrahedra
Robotics / vision	homogeneous transforms; watch \(\det\) of rotation blocks \(=1\)
Optimization	Newton steps solve linear systems, not inverses as matrices
Random matrices	volume of zonotopes; covariance geometry
Finite elements	Jacobian of reference maps

11. Pitfalls

1. \(\det(A+B)\neq\det A+\det B\)
   
2. \(\det(cA)=c^n\det A\), not \(c\det A\)
   
3. Using Cramer for large systems
   
4. Interpreting \(\det\approx 0\) without scaling context (scale rows by \(10^{10}\) inflates det)
   
5. Forming inverses “because the formula has \(A^{-1}\)”

12. Checkpoint

• Compute \(2\times 2\) and triangular determinants
  
• Use row-operation rules
  
• Apply \(\det(AB)=\det A\det B\)
  
• Interpret \(|\det|\) as volume scaling
  
• Explain why solvers beat explicit inverses

Exercises

Easy

1. Compute \(\det\begin{bmatrix}1&2&0\\0&3&4\\0&0&5\end{bmatrix}\).
   
2. Show that if two columns are equal, \(\det=0\).
   
3. If \(\det A=2\) and \(\det B=-3\), find \(\det(A^2 B^{-1})\) (\(B\) invertible).
   
4. For shear \(\begin{bmatrix}1&k\\0&1\end{bmatrix}\), compute det and interpret area.
   
5. Invert \(\begin{bmatrix}1&2\\3&4\end{bmatrix}\) if possible.

Medium

6. Explain why solving \(Ax=b\) via \(x=A^{-1}b\) is usually a bad numerical plan.
   
7. Prove \(\det(A^\top)=\det A\) for \(2\times 2\) by expansion.
   
8. Using only row ops, compute \(\det\begin{bmatrix}0&1&2\\3&4&5\\6&7&8\end{bmatrix}\).
   
9. If \(Q\) orthogonal, show \(|\det Q|=1\).
   
10. Relate \(\det A\) to the product of eigenvalues for a diagonalizable \(A\).

Challenge

11. Prove \(\det(AB)=\det A\det B\) for \(2\times 2\) by direct expansion.
    
12. Show \(\det(e^B)=e^{\mathrm{tr}(B)}\) for square \(B\) (use eigenvalues or series intuition).
    
13. Hadamard inequality: \(|\det A|\le\prod_j \|a_j\|_2\) for columns \(a_j\) — interpret geometrically.
    
14. Cramer’s rule derivation from \(A\,\mathrm{adj}(A)=(\det A)I\).
    
15. Floating point: construct a matrix with huge condition number but moderate entries; compare \(\det\) vs reliable rank-revealing QR.

Checks

1. \(15\).
   
2. \(4\cdot(-1/3)=-4/3\).
   
3. det \(=1\), area preserved.
   
4. \(\det=-2\), inverse \(-\frac12\begin{bmatrix}4&-2\\-3&1\end{bmatrix}\).

Summary

Determinants package volume, orientation, and invertibility into one scalar. Compute them via small expansions, pivots, or library routines; reason with multiplicativity and eigenvalue products. Inverses exist exactly when \(\det\neq 0\), but the mature computational habit is: solve systems, factor matrices, monitor conditioning — not chase giant explicit inverses. Geometry (shears, orthogonal maps) and applications (Jacobians, Gaussians) keep the scalar meaningful beyond algebra drills.