Lesson 4 Inverses and Determinants
4.2 Lecture Notes
4.2.1 Determinant of a \(2 \times 2\) Matrix
- For a matrix: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
- The determinant is: \[ \text{det}(A) = ad - bc \]
4.2.2 Determinant of a \(3 \times 3\) Matrix
- For a matrix: \[ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]
- Use the cofactor expansion along the first row: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
4.2.3 Determinant of Higher-Order Matrices
- Choose a row or column (often the one with the most zeros).
- Expand using cofactors:
- For each element \(a_{ij}\) in the row/column:
- Compute the minor (the determinant of the submatrix that remains when row \(i\) and column \(j\) are removed).
- Multiply by \((-1)^{i+j}\) (the sign factor).
- For each element \(a_{ij}\) in the row/column:
- Sum all the signed minors: \[ \text{det}(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij} \]
4.2.4 Inverse of a \(2 \times 2\) Matrix
- For: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
- If \(\text{det}(A) \neq 0\), the inverse is: \[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
4.2.5 Inverse of Higher-Order Matrices (Steps)
- Check that \(\text{det}(A) \neq 0\) (only non-singular matrices have inverses).
- Augment the matrix \(A\) with the identity matrix \(I\): \[ [A | I] \]
- Use Gauss-Jordan elimination to reduce \(A\) to the identity matrix.
- When \(A\) becomes \(I\), the right-hand side becomes \(A^{-1}\): \[ [I | A^{-1}] \]
4.3 Examples
Find the determinants, if they exist.
- \[
\begin{bmatrix}
3 & 1 \\
1 & 5
\end{bmatrix}
\]
- \[
\begin{bmatrix}
1 & -1 & 1 \\
-4 & 5 & -2
\end{bmatrix}
\]
- \[ \begin{bmatrix} -3 & 1 & 1 \\ -2 & -5 & 1 \\ -5 & -3 & -1 \end{bmatrix} \]
- \[
\begin{bmatrix}
3 & 1 \\
1 & 5
\end{bmatrix}
\]
Use the shortcut for finding the inverse of a \(2 \times 2\) matrix to find the inverse of:
\[ \begin{bmatrix} 4 & 0 \\ 2 & 1 \end{bmatrix} \]
True or False: A square matrix \(M\) is singular if and only if \(\det(M) \neq 0\).
Find the determinant of the matrix:
\[ \begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix} \]
- Find the determinant of the matrix:
\[ \begin{bmatrix} -2 & 5 \\ -1 & 4 \end{bmatrix} \]
- Find the inverse of the matrix using row operations:
\[ \begin{bmatrix} 6 & -2 \\ -2 & 4 \end{bmatrix} \]
- Does the matrix
\[ A = \begin{bmatrix} -4 & 13 \\ 4 & -13 \end{bmatrix} \] have an inverse?
- Find the inverse of the matrix:
\[ \begin{bmatrix} 7 & -2 \\ 21 & -6 \end{bmatrix} \]
- Find the determinant of the matrix:
\[ \begin{bmatrix} 3 & 0 & 0 \\ 7 & 4 & 0 \\ 2 & -1 & 6 \end{bmatrix} \]
- Find the determinant, if it exists:
\[ A = \begin{bmatrix} -3 & 0 & 1 \\ 4 & -3 & 0 \\ 4 & -3 & 0 \end{bmatrix} \]
4.4 Practice Problems
Practice the techniques discussed in class and in the online videos by solving the following examples.
- Calculate the determinant of \(\begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}\)
- Calculate the determinant of \(\begin{bmatrix} 0 & 1 & 2 \\ 3 & 1 & 0 \\ 1 & 1 & -1 \end{bmatrix}\)
- Find the determinant of \(\begin{bmatrix} 1 & 2 & 3 & 16 \\ 0 & 2 & 6 & -7 \\ 0 & 0 & 3 & 33 \\ 0 & 0 & 0 & -1 \end{bmatrix}\)
- If possible, find the inverses of the following matrices:
- \(\begin{bmatrix} 2 & 4 \\ -1 & 6 \end{bmatrix}\)
- \(\begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}\)
- Determine if \(A = \begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 & 3 \\ 5 & 1 \end{bmatrix}\) are invertible, and find their inverses.
- Solve the following system using a matrix inverse: \[ \begin{align*} \frac{2}{3}x - \frac{1}{2}y &= \frac{1}{6} \\ \frac{1}{2}x - \frac{1}{2}y &= -1 \end{align*} \]
4.5 Self-Assessment
Time yourself and try to solve the following questions within twenty minutes.
- If \(A(X + B) = C\), where \(A\), \(B\), \(X\), and \(C\) are all \(2 \times 2\) invertible matrices, solve for \(X\).
- Calculate the determinant of \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\)
- Find the inverse of \(\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\)
- Find the inverse of \(\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\)
- Solve the system using a matrix inverse: \[ \begin{align*} 2x + y &= 2 \\ 2x - 3y &= 2 \end{align*} \]
4.6 Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.
\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]
| Skill | D | CON | COM |
|---|---|---|---|
| Find the determinant and inverse of a \(2 \times 2\) matrix | |||
| Find the determinant and inverse of a \(3 \times 3\) matrix | |||
| Find inverses and determinants for higher-order matrices | |||
| Solve a linear system using a matrix inverse |