Lab 6 Interactive: Determinants

1. A $2\times2$ matrix as an area machine

Enter a matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$. The columns determine the image of the unit square.

2. Geometry of $A$

Blue: original unit square. Orange: transformed parallelogram. The orange area is $|\det(A)|$.

3. Compute a $3\times3$ determinant

Use this to compare direct determinant computation with elimination ideas.

4. Row-operation interpretation

Row replacement does not change determinant. Row swaps change sign. Row scaling multiplies determinant.

5. Eigenvalues and determinant

For triangular matrices, eigenvalues are the diagonal entries, so the determinant is their product.

$\lambda_1$$\lambda_2$$\lambda_3$

6. Singular values and condition number

For singular values $\sigma_1\ge\sigma_2>0$, $|\det(A)|=\sigma_1\sigma_2$ and $\kappa_2(A)=\sigma_1/\sigma_2$.

$\sigma_1$$\sigma_2$

7. Covariance volume

For a covariance matrix $\Sigma$, $\det(\Sigma)$ is generalized variance and $\sqrt{\det(\Sigma)}$ is proportional to ellipse volume.

$v_1$$v_2$$\rho$

8. Log determinant

For positive definite matrices, log determinant is more stable than determinant.

9. Independent-study practice tasks with answers

Task A. Compute $\det\begin{bmatrix}2&1\\0&3\end{bmatrix}$.
Answer: $6$. The map scales area by $6$ and preserves orientation.

Task B. Compute $\det\begin{bmatrix}1&0\\0&-1\end{bmatrix}$.
Answer: $-1$. The map preserves area but reverses orientation.

Task C. Decide whether $\begin{bmatrix}1&2\\2&4\end{bmatrix}$ is invertible.
Answer: No. Its determinant is $0$ and its columns are dependent.

Task D. If a $4\times4$ matrix has eigenvalues $2,-1,3,5$, find its determinant.
Answer: $2(-1)(3)(5)=-30$.

Similar practice. If the singular values are $10,1,0.01$, find $|\det(A)|$ and $\kappa_2(A)$.
Answer: $|\det(A)|=0.1$ and $\kappa_2(A)=1000$.