Elementary matrix

$Elementary matrix$

1. (1) P_{i, j} (c) = E + c E_{i, j} = ⎝ ⎛ 1 ⋱ 1 \dots ⋱ c ⋮ 1 ⋱ 1 ⎠ ⎞ 2. Q_{i, j} = ⎝ ⎛ 1 ⋱ 1 0 ⋮ 1 \dots \dots 1 ⋮ 0 1 ⋱ 1 ⎠ ⎞ 3. R_{i} (d) = ⎝ ⎛ 1 ⋱ 1 d 1 ⋱ 1 ⎠ ⎞

1. (P_{i, j} (c))^{- 1} = P_{i, j} (- c) 2. (Q_{i, j})^{- 1} = Q_{i, j} 3. (R_{i} (d))^{- 1} = R_{i} (d^{- 1})

The number of linearly independent vectors is preserved even after a matrix transformation.

\tilde{A} = (A E_{n}) = ⎝ ⎛ a_{11} ⋮ ⋮ a_{n 1} \dots \dots \dots \dots a_{1 n} ⋮ ⋮ a_{nn} 10 ⋮ 0 0 ⋱ ⋱ \dots \dots ⋱ ⋱ 0 0 ⋮ 01 ⎠ ⎞

A^{- 1} \tilde{A} = ((A^{- 1} A) (A^{- 1} E_{n})) = (E_{n} A^{- 1})

(E_{n} X) \Rightarrow A^{- 1} = X

A regular matrix

A

becomes a unit matrix by repeating the transformation by an elementary matrix