Theorems and Definitions in Linear Algebra

Determinants

$A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$

is a 2×2 matrix with entries form a field F, then we define the determinant of A, denoted det(A) or |A|, to be the scalar .

＊Theorem 1: linear function for a single row.
＊Theorem 2: nonzero determinant ⇔ invertible matrix

Theorem 1: The function det: M_2×2(F) → F is a linear function of each row of a 2×2 matrix when the other row is held fixed. That is, if and are in F² and is a scalar, then

$\det\begin{pmatrix} u + kv\\ w\\ \end{pmatrix} =\det\begin{pmatrix} u\\ w\\ \end{pmatrix} + k\det\begin{pmatrix} v\\ w\\ \end{pmatrix}$

and

$\det\begin{pmatrix} w\\ u + kv\\ \end{pmatrix} =\det\begin{pmatrix} w\\ u\\ \end{pmatrix} + k\det\begin{pmatrix} w\\ v\\ \end{pmatrix}$

Theorem 2: Let A M_2×2(F). Then thee deter minant of A is nonzero if and only if A is invertible. Moreover, if A is invertible, then

$A^{-1}=\frac{1}{\det(A)}\begin{pmatrix} A_{22}&-A_{12}\\ -A_{21}&A_{11}\\ \end{pmatrix}$

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Theorems and Definitions in Linear Algebra - Determinants