1. Interchange any two equations
2. Multiply an equation by a non-zero constant
3. Add an equation to another equation

1. \(\frac{x + 7}{x^2 - x - 6} = \frac{x + 7}{(x - 3)(x + 2)}\)

\[ \frac{A}{x - 3} + \frac{B}{x + 2} \]

\[ (\frac{x + 7}{(x - 3)(x + 2)} = \frac{A}{x - 3} + \frac{B}{x + 2})(x + 2) \]

\[ \frac{x + 7}{x - 3} = \frac{A(x + 2)}{x - 3} + B \]

Let \(x = -2\)

\[ B = \frac{-2 + 7}{-2 + 3} = 5 \]

2. \(\frac{x - 3}{x^2 - 16}\)

\[ \frac{x - 3}{(x + 4)(x - 4)} \]

\[ \frac{} \]

When multiplying matrices \(A \cdot B = C\), an element \(e\) at \(C_{y,x}\) is the sum of the products of \(A\) at row \(y\) by its corresponding elements \(B\) at column \(x\).

• Two types of inverses in real numbers:

  • Additive inverse, called the opposite, \(x + y = 0\)
  • Multiplicative inverse, called the reciprocal, \(x \cdot y = 1\)

\[ \begin{bmatrix} A \end{bmatrix} \cdot \begin{bmatrix} A \end{bmatrix}^{-1} = \begin{bmatrix} I \end{bmatrix} \]

• Identity matrix contains all ones in the main diagonal
• All other elements must be zero

• Not all matrices have inverses

  • Only square matrices are capable of having inverses
  • Square matrices that do not have inverses are known as singular or non-invertible

Given that \(AX = I\) where \(X = A^{-1}\)

\[ \begin{bmatrix} A \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ -1 & -3 \end{bmatrix}, \]

\[ \left[\begin{array}{rr|rr} 1 & 4 & 1 & 0 \\ -1 & -3 & 0 & 1 \end{array}\right] \]

\[ \left[\begin{array}{rr|rr} 1 & 4 & 1 & 0 \\ 0 & 1 & 1 & 1 \end{array}\right] \]

\[ \left[\begin{array}{rr|rr} 1 & 0 & -3 & -4 \\ 0 & 1 & 1 & 1 \end{array}\right] \]

What we have on the right hand side is the matrix \(\begin{bmatrix} X \end{bmatrix}\)

\[ \left[\begin{array}{cc|cc} a & b & 1 & 0 \\ c & d & 0 & 1 \end{array}\right] \]

\[ \left[\begin{array}{cc|cc} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ c & d & 0 & 1 \end{array}\right] \]

\[ \left[\begin{array}{cc|cc} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ 0 & \frac{ad - bc}{a} & -\frac{c}{a} & 1 \end{array}\right] \]

\[ \left[\begin{array}{cc|cc} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ 0 & 1 & -\frac{c}{ad - bc} & \frac{a}{ad - bc} \end{array}\right] \]

\[ \left[\begin{array}{cc|cc} 1 & 0 & \frac{d}{ad - bc} & -\frac{b}{ad - bc} \\ 0 & 1 & -\frac{c}{ad - bc} & \frac{a}{ad - bc} \end{array}\right] \]

\[ \frac{1}{ad - bc} \left[\begin{array}{rr} d & -b \\ -c & a \end{array}\right] \]

An equation

\[ \begin{array}{c} x + 2y = 4 \\ 3x - 5y = 1 \end{array} \]

can be written in the form

\[ \left[\begin{array}{cc} 1 & 2 \\ 3 & -5 \end{array}\right] \left[\begin{array}{c} x \\ y \end{array}\right] = \left[\begin{array}{c} 4 \\ 1 \end{array}\right] \]

The second matrix \(B = A^{-1}C\). Therefore, \[ B = -\frac{1}{11} \left[\begin{array}{cc} -5 & -2 \\ -3 & 1 \end{array}\right] \left[\begin{array}{c} 4 \\ 1 \end{array}\right] \\ = -\frac{1}{11} \left[\begin{array}{cc} -22 \\ -11 \end{array}\right] = \left[\begin{array}{cc} 2 \\ 1 \end{array}\right] \]

\[ \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right] \]

Minor

\[ a_{11} \left[\begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array}\right] \]

3x3 determinant

\[ \left[\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \]

\[ a(ei - fh) - b(di - fg) + c(dh - eg) = (aei + bfg + cdh) - (afh - bdi + ceg) \]

\[ A = ≤ft[\begin{array}{ccc} 4 & -3 & 0 \\ 2 & -1 & 2 \\ 1 & 5 & 7 \end{array}\right],

\]

\[ |A| = -32 \]