Last modified on 31 May 2010, at 18:24

Linear Algebra/Definition of Determinant

Linear Algebra
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For  1 \! \times \! 1 matrices, determining nonsingularity is trivial.

 \begin{pmatrix}
a
\end{pmatrix}  is nonsingular iff  a \neq 0

The 2 \! \times \! 2 formula came out in the course of developing the inverse.

 \begin{pmatrix}
a  &b  \\
c  &d
\end{pmatrix}  is nonsingular iff  ad-bc \neq 0

The 3 \! \times \! 3 formula can be produced similarly (see Problem 9).

 \begin{pmatrix}
a  &b  &c  \\
d  &e  &f  \\
g  &h  &i
\end{pmatrix} is nonsingular iff  aei+bfg+cdh-hfa-idb-gec \neq 0

With these cases in mind, we posit a family of formulas, a, ad-bc, etc. For each n the formula gives rise to a determinant function \det\nolimits_{n \! \times \! n}:\mathcal{M}_{n \! \times \! n}\to \mathbb{R} such that an n \! \times \! n matrix T is nonsingular if and only if \det\nolimits_{n \! \times \! n}(T)\neq 0. (We usually omit the subscript because if  T is  n \! \times \! n then " \det(T) " could only mean " \det\nolimits_{n \! \times \! n}(T) ".)


Linear Algebra
 ← Determinants Definition of Determinant Exploration →