Lab 6. Determinants: Independent Study
This lab accompanies Chapter 6: Determinants.
The goal is to connect the story of determinants with computation:
- Geometry: determinant as signed area or volume scaling.
- Invertibility: nonzero determinant means no dimension is lost.
- Row operations: compute determinants efficiently by elimination.
- Eigenvalues and singular values: determinant as a product of spectral quantities.
- Applications: covariance volume, condition number, and log determinant.
This is an independent-study lab. Each main question includes a worked solution and a similar practice question.
Python practice notebook
You may also use the Jupyter notebook version for longer Python practice:
Interactive lab
Study guide and worked questions
Question 1. Area scaling in \(\mathbb R^2\)
Let
\[ A=\begin{bmatrix}2&1\\0&3\end{bmatrix}. \]
Compute \(\det(A)\) and interpret the answer geometrically.
Solution
\[ \det(A)=2\cdot3-1\cdot0=6. \]
The transformation \(x\mapsto Ax\) sends the unit square to a parallelogram of area \(6\). Since the determinant is positive, orientation is preserved.
Similar practice
Let
\[ B=\begin{bmatrix}1&4\\0&1\end{bmatrix}. \]
Compute \(\det(B)\) and explain why this shear preserves area.
Answer
\[ \det(B)=1\cdot1-4\cdot0=1. \]
It preserves area but changes the shape of the unit square.
Question 2. Orientation
Let
\[ R=\begin{bmatrix}1&0\\0&-1\end{bmatrix}. \]
Compute \(\det(R)\) and interpret the sign.
Solution
\[ \det(R)=1(-1)-0=-1. \]
The transformation reflects across the \(x\)-axis. It preserves area but reverses orientation.
Similar practice
Compute the determinant of
\[ Q=\begin{bmatrix}0&-1\\1&0\end{bmatrix}. \]
Answer
\[ \det(Q)=0\cdot0-(-1)\cdot1=1. \]
This rotation preserves area and orientation.
Question 3. Compute a \(3\times3\) determinant by row operations
Let
\[ A=\begin{bmatrix} 1&2&3\\ 2&5&7\\ 1&1&0 \end{bmatrix}. \]
Compute \(\det(A)\) using elimination.
Solution
Use row replacements, which do not change the determinant:
\[ R_2\leftarrow R_2-2R_1, \qquad R_3\leftarrow R_3-R_1. \]
This gives
\[ \begin{bmatrix} 1&2&3\\ 0&1&1\\ 0&-1&-3 \end{bmatrix}. \]
Then use
\[ R_3\leftarrow R_3+R_2, \]
which gives
\[ \begin{bmatrix} 1&2&3\\ 0&1&1\\ 0&0&-2 \end{bmatrix}. \]
Therefore
\[ \det(A)=1\cdot1\cdot(-2)=-2. \]
Similar practice
Compute the determinant of
\[ B=\begin{bmatrix} 2&1&0\\ 1&3&1\\ 0&2&4 \end{bmatrix}. \]
Answer
Expanding along the first row or using elimination gives
\[ \det(B)=2(3\cdot4-1\cdot2)-1(1\cdot4-1\cdot0)=2(10)-4=16. \]
Question 4. Determinant and invertibility
Let
\[ A=\begin{bmatrix}1&2\\2&4\end{bmatrix}. \]
Determine whether \(A\) is invertible.
Solution
\[ \det(A)=1\cdot4-2\cdot2=0. \]
Therefore \(A\) is not invertible. The columns are linearly dependent because the second column is twice the first column.
Similar practice
Let
\[ B=\begin{bmatrix}3&1\\2&5\end{bmatrix}. \]
Determine whether \(B\) is invertible.
Answer
\[ \det(B)=15-2=13\ne0. \]
So \(B\) is invertible.
Question 5. Determinant from eigenvalues and singular values
Suppose a \(4\times4\) matrix has eigenvalues
\[ 2,\quad -1,\quad 3,\quad 5. \]
Find its determinant.
Solution
\[ \det(A)=2(-1)(3)(5)=-30. \]
Since the determinant is nonzero, \(A\) is invertible.
Similar practice
Suppose a \(3\times3\) matrix has singular values
\[ 10, \quad 1, \quad 0.01. \]
Find \(|\det(A)|\) and the condition number.
Answer
\[ |\det(A)|=10\cdot1\cdot0.01=0.1. \]
The condition number is
\[ \kappa_2(A)=\frac{10}{0.01}=1000. \]
Question 6. Covariance volume
Let
\[ \Sigma=\begin{bmatrix}4&0\\0&1\end{bmatrix}. \]
Compute \(\det(\Sigma)\) and interpret it as generalized variance.
Solution
\[ \det(\Sigma)=4. \]
The generalized variance is \(4\). The covariance ellipse has area proportional to
\[ \sqrt{\det(\Sigma)}=2. \]
Similar practice
Let
\[ \Sigma=\begin{bmatrix}9&0\\0&4\end{bmatrix}. \]
Compute the generalized variance.
Answer
\[ \det(\Sigma)=36. \]
The covariance ellipse has area proportional to \(6\).
AI companion activities
Activity 1. Check one determinant in three ways
Ask an AI assistant:
Compute \(\det\begin{bmatrix}1&2&3\\2&5&7\\1&1&0\end{bmatrix}\) by cofactor expansion, row reduction, and Python. Which method is most efficient?
Then compare the answer with this lab.
Activity 2. Explain determinant zero
Ask:
Explain geometrically why \(\det(A)=0\) means a square matrix loses dimension.
Check whether the answer mentions linear dependence of columns.
Activity 3. Condition number warning
Ask:
Give a matrix with determinant \(1\) but very large condition number. Explain why determinant alone is not enough for numerical stability.
Then compute its singular values in Python.
Activity 4. Gaussian connection
Ask:
Why does the determinant of a covariance matrix appear in the multivariate Gaussian density?
Summarize the answer using the phrase volume normalization.