1What is the method for computing right inverses of non-square matrices? ($A_{m \times n}$)
1. Create augmented matrix $[A \mid I_m]$
2. Row reduce augmented matrix to get $\text{RREF}([A \mid I_m]) = [I_m \mid B]$
3. B is a right inverse of A
2What is the method for computing left inverses of non-square matrices? ($A_{m \times n}$)
1. Create augmented matrix $\left[ A^T \mid I_n \right]$
2. Row reduce augmented matrix to get $\text{RREF}([A^T \mid I_n]) = [I_n \mid B]$
3. $B^T$ is a left inverse of A
3What is the method for computing inverses of square matrices? ($A_{m \times m}$)
1. Create augmented matrix $\left[ A \mid I_m \right]$
2. Row reduce augmented matrix to get $\text{RREF}([A \mid I_m]) = [I_m \mid B]$
3. $B = A^{-1}$
4What is the method for computing the determinant of a 2x2 matrix?
$A = \begin{bmatrix} a & \beta \\ \gamma & \delta \end{bmatrix}$
$\det(A) = a\delta - \beta\gamma$
5What is the method for computing the determinant of a 3x3 matrix?
$A = \begin{bmatrix}
\alpha & \beta & \delta \\
\gamma & \lambda & \nu \\
\epsilon & \pi & \tau
\end{bmatrix}$
$\det(A) = +\alpha \begin{vmatrix} \lambda & \nu \\ \pi & \tau \end{vmatrix} - \beta \begin{vmatrix} \gamma & \nu \\ \epsilon & \tau \end{vmatrix} + \delta \begin{vmatrix} \gamma & \lambda \\ \epsilon & \pi \end{vmatrix}$
6What is the shortcut for the determinant of a diagonal, uppper/lower triangular matrix?
det(A) = product of diag(A)
7What is det(B) if B is obtained from A via the ERO $(r_i \to r_i + \lambda r_j)$?
det(B) = det(A)
8$\text{What is the adjugate of } A \in M_{3 \times 3}?$
$\text{adj} \left(
\begin{bmatrix}
\alpha & \beta & \gamma \\
\delta & \lambda & \mu \\
\epsilon & \pi & \tau
\end{bmatrix}
\right) =
\begin{bmatrix}
\begin{vmatrix}
\lambda & \mu \\
\pi & \tau
\end{vmatrix} &
\begin{vmatrix}
\delta & \mu \\
\epsilon & \tau
\end{vmatrix} &
\begin{vmatrix}
\delta & \lambda \\
\epsilon & \pi
\end{vmatrix} \\
\begin{vmatrix}
\beta & \gamma \\
\pi & \tau
\end{vmatrix} &
\begin{vmatrix}
\alpha & \gamma \\
\epsilon & \tau
\end{vmatrix} &
\begin{vmatrix}
\alpha & \beta \\
\epsilon & \pi
\end{vmatrix} \\
\begin{vmatrix}
\beta & \gamma \\
\lambda & \mu
\end{vmatrix} &
\begin{vmatrix}
\alpha & \gamma \\
\delta & \mu
\end{vmatrix} &
\begin{vmatrix}
\alpha & \beta \\
\delta & \lambda
\end{vmatrix}
\end{bmatrix}^T
\cdot
\begin{bmatrix}
+ & - & + \\
- & + & - \\
+ & - & +
\end{bmatrix}$
9$\text{How is the inverse of } A \text{ related too adj(A)?}$
$A^{-1} = \frac{\text{adj}(A)}{\det(A)}$