📐 Invertibility and Determinant

A square matrix $A$ is invertible if and only if $\\det(A) \\neq 0$.

Proof: **(⇒)** If $A$ is invertible, then $AA^{-1} = I$. By the product rule:

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

So $\\det(A) \\neq 0$.

**(⇐)** If $\\det(A) \\neq 0$, then the columns of $A$ are linearly independent (if dependent, two columns would be proportional, giving det = 0). Thus $A$ has f...

From: Advanced Linear Algebra

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