Math Notes

November 9, 2015

Fractions.

  • Operations

$$ \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd} $$ $$ \frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd} $$ $$ \frac{a}{b} x \frac{c}{d} = \frac{ac}{bd} $$ $$ \frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} $$

  • Inverse of an Inverse:

$$ (a^{-1})^{-1} = \left(\frac{1}{a}\right)^1 = a $$ $$ (a^{-1})^{-1} = a $$

  • Inverse of a fraction

$$ \frac{\frac{1}{a}}{b} = \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} $$

Matrices.

  • Determinants of 2x2 matrices are defined by:

  • The determinant of a 3 × 3 matrix is defined by

\begin{align}\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix} & = a\begin{vmatrix}e&f\\h&i\end{vmatrix}-b\begin{vmatrix}d&f\\g&i\end{vmatrix}+c\begin{vmatrix}d&e\\g&h\end{vmatrix} \\
& = a(ei-fh)-b(di-fg)+c(dh-eg) \\
& = aei+bfg+cdh-ceg-bdi-afh.
\end{align}

 Properties

$$ det(I) = 1 $$. $$ det( A^{T} )=det( A ) $$ $$ det(A^{-1}) = \frac {1}{det(A)} = det(A)^{-1} $$

for a square matrix A and B of equal size:

$$ det(AB)= det(A) det(B) $$ $$ det(cA)= c^{n} det(A) $$ for a square matrix.

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