Section 2 Theorems

Theorem 1

  • If A is an invertible Matrix then $A^{-1}$ is invertible
  • If A and B are invertible, then AB is invertible

$(AB)(B^{-1}A^{-1}) = I = AA^{-1} = I$

  • If A in invertible then $A^T$ is invertible

If

(1)
\begin{align} A = \begin{bmatrix} a&b&|&1&0 \\ c&d&|&0&1 \end{bmatrix} \longrightarrow \begin{bmatrix} 1&0&|&e&f\\0&1&|&g&h \end{bmatrix} \end{align}

then

(2)
\begin{align} B = \begin{bmatrix} a&c&|&1&0 \\ b&d&|&0&1 \end{bmatrix} \longrightarrow \begin{bmatrix} 1&0&|&e&g \\ 0&1&|&f&h \end{bmatrix} \end{align}

must also be true because

(3)
\begin{align} \begin{bmatrix} a&b\\c&d \end{bmatrix}^{-T} = \begin{bmatrix} a&b\\c&d \end{bmatrix}^{T(-1)} \end{align}

and we can transpose $A^{-1}$

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