[Linear Algebra] Non Invertible Matrix

Non-Invertible Matrix $A$ for $A \textbf{x} = \textbf{b}$

• If matrix $A$ is invertible, the solution is uniquely obtained as $\textbf{x} = A^{-1} \textbf{b}$
• If matrix $A$ is non-invertible, $A \textbf{x} = \textbf{b}$ will have either no solution of infinitely many solutions

Does a Matrix Have an Inverse Matrix?

• $\textrm{det} A$ determines whether matrix $A$ is invertible or not
  • if $\textrm{det} A = 0$, non-invertible
  • if $\textrm{det} A \neq 0$, invertible

Properties of Determinants

Defining Properties

1. $\textrm{det} I_n = 1$

2. Exchanging rows reverse the sign of $\textrm{det}$

   e.g.
   $A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$, $B = \begin{bmatrix} c & d \\ a & b \\ \end{bmatrix}$ then $\textrm{det} A = -\textrm{det} B$

3. Linearity in each rows

   e.g. $\textrm{det} C = c * \textrm{det} A$

   $A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$, $C = \begin{bmatrix} ca & cb \\ c & d \\ \end{bmatrix}$

   e.g. $\textrm{det} C = \textrm{det} A + \textrm{det} B$

   $A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$, $B = \begin{bmatrix} a' & b' \\ c & d \\ \end{bmatrix}$, $C = \begin{bmatrix} a+a' & b+b' \\ c & d \\ \end{bmatrix}$

   • Caution : this does not mean $\textrm{det} (A + B) = \textrm{det} A + \textrm{det} B$, which is wrong

Properties that are Proved from the Defining Properties

4. If two rows of matrix $A$ are equal, then $\textrm{det} A = 0$

   • use property 2

     e.g.

     $\begin{vmatrix} a & b & c \\ d & e & f \\ a & b & c \\ \end{vmatrix} = -\begin{vmatrix} a & b & c \\ d & e & f \\ a & b & c \\ \end{vmatrix} = 0$

5. Subtracting some rows with certain row doesn't change determinant

   • use property 3 & 4

     e.g.

     $\begin{vmatrix} a+kc & b+kd \\ c & d \\ \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} + \begin{vmatrix} kc & kd \\ c & d \\ \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} + k \begin{vmatrix} c & d \\ c & d \\ \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}$

6. If matrix $A$ has row of zeros, then $\textrm{det} A = 0$

   • use property 3

     e.g.

     $\begin{vmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 0 \\ \end{vmatrix} = \begin{vmatrix} a & b & c \\ d & e & f \\ k \times 0 & k \times 0 & k \times 0 \\ \end{vmatrix} = k \begin{vmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 0 \\ \end{vmatrix} = 0$

7. If matrix $A$ is upper triangular of lower triangular, then $\textrm{det} A = a_{11} a_{22} \cdots a_{nn}$, product of the diagonal

   • use property 1 & 3 & 5

8. $\textrm{det} A = 0$ when matrix $A$ is singular and $\textrm{det} A \neq 0$ when matrix $A$ is invertible

9. If matrix $A$ and $B$ are $n \times n$ matrices, $\textrm{det} (AB) = (\textrm{det} A) (\textrm{det} B)$

   • $\textrm{det} A^{-1} = \frac{1}{\textrm{det} A}$

10. $\textrm{det} A^T = \textrm{det} A$

Determinant Formulas and Cofactors

Formula for the Determinant

• Just using the properties of determinants, we can get the formula of determinant

$\textrm{det} A = \sum_{n! terms} \pm a_{1, \alpha} a_{2, \beta} a_{3, \gamma} \cdots a_{n, \omega}$

where $(\alpha, \beta, \gamma, \cdots, \omega)$ is some permutation of $(1, 2, 3, \cdots, n)$

e.g. 2 x 2 matrix

$\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = \begin{vmatrix} a & 0 \\ c & d \\ \end{vmatrix} + \begin{vmatrix} 0 & b \\ c & d \\ \end{vmatrix} = \begin{vmatrix} a & 0 \\ c & 0 \\ \end{vmatrix} + \begin{vmatrix} a & 0 \\ 0 & d \\ \end{vmatrix} + \begin{vmatrix} 0 & b \\ c & 0 \\ \end{vmatrix} + \begin{vmatrix} 0 & b \\ 0 & d \\ \end{vmatrix}$

$= 0 + ad - bc + 0 = ad-bc$

e.g. 3 x 3 matrix

$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{vmatrix}$

$= \begin{vmatrix} a & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & i \\ \end{vmatrix} + \begin{vmatrix} a & 0 & 0 \\ 0 & 0 & f \\ 0 & h & 0 \\ \end{vmatrix} + \begin{vmatrix} 0 & b & 0 \\ d & 0 & 0 \\ 0 & 0 & i \\ \end{vmatrix} + \begin{vmatrix} 0 & b & 0 \\ 0 & 0 & f \\ g & 0 & 0 \\ \end{vmatrix} + \begin{vmatrix} 0 & 0 & c \\ d & 0 & 0 \\ 0 & h & 0 \\ \end{vmatrix} + \begin{vmatrix} 0 & 0 & c \\ 0 & e & 0 \\ g & 0 & 0 \\ \end{vmatrix}$

$= aei - afh - bdi + bfg + cdh - ceg$

Cofactor Formula

• The cofactor formula rewrites the big formula for the determinant of and $n \times n$ matrix in terms of the determinants of smaller matrices
• Cofactor of $a_{ij}$ = $C_{ij}$ = $\pm \textrm{det}$ of $n-1$ matrix with row $i$, column $j$ erased
  • if $i + j$ is even, the sign is $+$
  • if $i + j$ is odd, the sign is $-$

$\textrm{det} A = a_{11} C_{11} + a_{12} C_{12} + \cdots + a_{1n} C_{1n}$

e.g. 3 x 3 matrix

$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{vmatrix}$

$= \begin{vmatrix} a & 0 & 0 \\ 0 & e & f \\ 0 & h & i \\ \end{vmatrix} + \begin{vmatrix} 0 & b & 0 \\ d & 0 & f \\ g & 0 & i \\ \end{vmatrix} + \begin{vmatrix} 0 & 0 & c \\ d & e & 0 \\ g & h & 0 \\ \end{vmatrix}$

$= a(\textrm{det} \begin{vmatrix} e & f \\ h & i \\ \end{vmatrix}) + b(- \textrm{det} \begin{vmatrix} d & f \\ g & i \\ \end{vmatrix}) + c(\textrm{det} \begin{vmatrix} d & e \\ g & h \\ \end{vmatrix})$

$= a(ei - fh) + b(-di + fg) + c(dh - eg)$

$= aei - afh - bdi + bfg + cdh - ceg$