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Question Number 174220 by Mastermind last updated on 27/Jul/22

Commented by MJS_new last updated on 27/Jul/22

$$-21$$

Commented by Mastermind last updated on 27/Jul/22

$$\mathrm{Show}\mathrm{workings}!$$$$$$

Commented by MJS_new last updated on 27/Jul/22

$$\mathrm{use}\mathrm{formula}$$

Commented by Mastermind last updated on 27/Jul/22

$$\mathrm{If}\mathrm{you}\mathrm{want}\mathrm{to}\mathrm{help},\mathrm{help}!$$$$\mathrm{If}\mathrm{we}\mathrm{can}\mathrm{do}\mathrm{it},\mathrm{i}{\mathrm{won}}^{\prime }\mathrm{t}\mathrm{posted}\mathrm{it}\mathrm{here}$$

Commented by MJS_new last updated on 27/Jul/22

$$\det \left[\begin{array}{c}M\end{array}\right]=\left|\begin{array}{c}M\end{array}\right|$$$$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$$$$$$$$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=aei+bfg+cdh-(afh+bdi+ceg)=$$$$=a(ei-fh)-b(di-fg)+c(dh-eg)=$$$$=a\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+c\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|$$$$$$$$\mathrm{same}\mathrm{principle}\mathrm{here}:$$$$\left|\begin{array}{cccc}a& b& c& d\\ e& f& g& h\\ i& j& k& l\\ m& n& o& p\end{array}\right|=a\left|\begin{array}{ccc}f& g& h\\ j& k& l\\ n& o& p\end{array}\right|-b\left|\begin{array}{ccc}e& g& h\\ i& k& l\\ m& o& p\end{array}\right|+c\left|\begin{array}{ccc}e& f& h\\ i& j& l\\ m& n& p\end{array}\right|-d\left|\begin{array}{ccc}e& f& g\\ i& j& k\\ m& n& o\end{array}\right|$$

Commented by Mastermind last updated on 27/Jul/22

$$\mathrm{Thanks}\mathrm{man}$$

Commented by MJS_new last updated on 28/Jul/22

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{welcome}.\mathrm{this}\mathrm{principle}\mathrm{works}\mathrm{for}\mathrm{any}$$$$\mathrm{matrix}\mathrm{size}n\times n$$

Answered by Rasheed.Sindhi last updated on 27/Jul/22

$$\mathrm{Using}\mathrm{Properties}$$$$\left|\begin{array}{cccc}2& 1& -2& 3\\ 3& 2& -1& 2\\ 3& 3& 2& -3\\ 0& 4& 3& 1\end{array}\right|$$$$\left|\begin{array}{cccc}-1& -1& -1& 1\\ 3& 2& -1& 2\\ 3& 3& 2& -3\\ 0& 4& 3& 1\end{array}\right|\mathrm{R}1-\mathrm{R}2$$$$\left|\begin{array}{cccc}0& 0& 0& 1\\ 5& 4& 1& 2\\ 0& 0& -1& -3\\ 1& 5& 4& 1\end{array}\right|\underset{\underset{\mathrm{C}3+\mathrm{C}4}{\mathrm{C}2+\mathrm{C}4}}{\mathrm{C}1+\mathrm{C}4}$$$$-\left(1\right)\left|\begin{array}{ccc}5& 4& 1\\ 0& 0& -1\\ 1& 5& 4\end{array}\right|$$$$-1(-(-1))\left|\begin{array}{ccc}5& 4& -3\\ 0& 0& 1\\ 1& 5& 2\end{array}\right|$$$$=-\left|\begin{array}{cc}5& 4\\ 1& 5\end{array}\right|$$$$-(5\times 5-4\times 1)=-21$$

Commented by Mastermind last updated on 27/Jul/22

$$\mathrm{Thank}\mathrm{you}\mathrm{so}\mathrm{much},\mathrm{God}\mathrm{bless}\mathrm{you}$$

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