Question

Solve : $\det \begin{pmatrix}1\:&\:2\:&\:3\:\\ 4\:&\:5\:&\:6\:\\ 7\:&\:8\:&\:9\end{pmatrix}$

Answer 1

$\bf{\det \begin{pmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{pmatrix}=0}$

$\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}$

$\det \begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}=a\cdot \det \begin{pmatrix}e&f\\ h&i\end{pmatrix}-b\cdot \det \begin{pmatrix}d&f\\ g&i\end{pmatrix}+c\cdot \det \begin{pmatrix}d&e\\ g&h\end{pmatrix}$

$=1\cdot \det \begin{pmatrix}5&6\\ 8&9\end{pmatrix}-2\cdot \det \begin{pmatrix}4&6\\ 7&9\end{pmatrix}+3\cdot \det \begin{pmatrix}4&5\\ 7&8\end{pmatrix}$

$\det \begin{pmatrix}5&6\\ 8&9\end{pmatrix}=-3$

$\det \begin{pmatrix}4&6\\ 7&9\end{pmatrix}=-6$

$\det \begin{pmatrix}4&5\\ 7&8\end{pmatrix}=-3$

$=1\cdot \left(-3\right)-2\left(-6\right)+3\left(-3\right)$

$1\cdot \left(-3\right)-2\left(-6\right)+3\left(-3\right)=0$

$=0$

Answer 2

$\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}:\quad \det \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}\:=\:ad-bc$

$=5\cdot \:9-6\cdot \:8$

$=1\cdot \left(-3\right)-2\left(-6\right)+3\left(-3\right)$

$=0$