Determinant

Note

For $A=[a]\in M_{1x1}(\mathbb{R})$ the determinant of A is

det(A) = det([a]) = a

and for $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in M_{2x2}(\mathbb{R})$ the determinant of A is

$det(A)=det(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right])=ad-bc$

A is invertible if and only if det(A) $\ne$ 0

Properties of Determinant §

[!Theorem 6.3.2]

If $A\in M_{nxn}(\mathbb{R})$ and $c\in \mathbb{R}$ then $det(cA)=c^{n}det(A)$

[!Theorem 6.3.4]

If $A,B\in M_{mxn}(\mathbb{R})$, then $det(AB)=det(A)det(B)$

[!Theorem 6.3.7]

Let $A\in M_{nxn}(\mathbb{R})$ be invertible. Then

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

[!Theorem 6.3.10]

Let $A\in M_{nxn}(\mathbb{R})$ Then $det(A^T)=det(A)$