Inverse of a Matrix
Contents |
Theorem
Let $\mathbf A = \begin{bmatrix}a\end{bmatrix}_n$ be an invertible square matrix of order $n$.
Let $\mathbf B = \begin{bmatrix}b\end{bmatrix}_n = \mathbf A^{-1}$ be the inverse of $\mathbf A$.
Then $\mathbf B$ is defined as:
- $b_{ij} = \dfrac {A_{ji}} {\det \mathbf A}$
where $A_{ji}$ is the cofactor of $a_{ji} \in \mathbf A$.
Corollary
A square matrix is invertible iff its determinant is not zero.
Proof
A messy old beast to prove, this. Anyone care to have a go?
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Proof of Corollary
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Sources
- Seth Warner: Modern Algebra (1965): $\S 29$
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.3$
- John F. Humphreys: A Course in Group Theory (1996): $\text{A}.2$: Theorem $\text{A}.9 \ (2)$
