નિશ્ચાયક left|begin{array}{ccc}2 & -3 & 5 6 & 0 & 4 1 & 5 &

Hindi

3 and 4 .Determinants and Matrices

medium

નિશ્ચાયક $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ ના ઘટકોના ઉપનિશ્ચાયક અને સહઅવયવ શોધો તથા ચકાસો કે $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}=0$

$0$

Solution

we have $\mathrm{M}_{11}=\left|\begin{array}{cc}0 & 4 \\ 5 & -7\end{array}\right|=0-20=-20 ; \mathrm{A}_{11}=(-1)^{1+1}(-20)=-20$

$\mathrm{M}_{12}=\left|\begin{array}{cc}6 & 4 \\ 1 & -7\end{array}\right|=-42-4=-46 ; \quad \mathrm{A}_{12}=(-1)^{1+2}(-46)=46$

$\mathrm{M}_{13}=\left|\begin{array}{ll}6 & 0 \\ 1 & 5\end{array}\right|=30-0=30 ; \quad \mathrm{A}_{13}=(-1)^{1+3}(30)=30$

$\mathrm{M}_{21}=\left|\begin{array}{cc}-3 & 5 \\ 5 & -7\end{array}\right|=21-25=-4 ; \quad \mathrm{A}_{21}=(-1)^{2+1}(-4)=4$
$\mathrm{M}_{22}=\left|\begin{array}{cc}2 & 5 \\ 1 & -7\end{array}\right|=-14-5=-19 ; \quad \mathrm{A}_{22}= (-1)^{2+2}(-19)=-19$

$\mathrm{M}_{23}=\left|\begin{array}{cc}2 & -3 \\ 1 & 5\end{array}\right|=10+3=13 ; \quad \quad \mathrm{A}_{23}=(-1)^{2 * 3}(13)=-13$

$\mathrm{M}_{31}=\left|\begin{array}{cc}-3 & 5 \\ 0 & 4\end{array}\right|=-12-0=-12 ; \quad \mathrm{A}_{31}=(-1)^{3+1}(-12)=-12$

$M_{32}=\left|\begin{array}{ll}2 & 5 \\ 6 & 4\end{array}\right|=8-30=-22 ; \quad \quad A_{32}=(-1)^{3+2}(-22)=22$

and $\mathrm{M}_{33}=\left|\begin{array}{cc}2 & -3 \\ 6 & 0\end{array}\right|=0+18=18 ; \quad \quad \mathrm{A}_{33}=(-1)^{3+3}(18)=18$

Now $\quad a_{11}=2, a_{12}=-3, a_{13}=5 ; \mathrm{A}_{31}=-12, \mathrm{A}_{32}=22, \mathrm{A}_{33}=18$

So $\quad a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}$

$=2(-12)+(-3)(22)+5(18)=-24-66+90=0$

Standard 12

Mathematics

નીચે આપેલા નિશ્ચાયકના પ્રત્યેક ઘટકના ઉપનિશ્ચાયક અને સહઅવયવ લખો : $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$

easy

નિશ્રાયક $\left| {\,\begin{array}{*{20}{c}}1&3&5&1\\2&3&4&2\\8&0&1&1\\0&2&1&1\end{array}\,} \right|$ માં ઘટક $'4'$ નો સહઅવયવ મેળવો.

easy

નિશ્ચાયક $\left|\begin{array}{rr}1 & -2 \\ 4 & 3\end{array}\right|$ ના બધા જ ઘટકના ઉપનિશ્ચાયક અને સહઅવયવ શોધો.

easy

જો ${\Delta _1} = \left| {\begin{array}{{20}{c}}
{{b^5}{c^6}\left( {{c^3} – {b^3}} \right)}&{{a^4}{c^6}\left( {{a^3} – {c^3}} \right)}&{{a^4}{b^5}\left( {{b^3} – {a^3}} \right)} \\
{{b^2}{c^3}\left( {{b^6} – {c^6}} \right)}&{a{c^3}\left( {{c^6} – {a^6}} \right)}&{a{b^2}\left( {{a^6} – {b^6}} \right)} \\
{{b^2}{c^3}\left( {{c^3} – {b^3}} \right)}&{a{c^3}\left( {{a^3} – {c^3}} \right)}&{a{b^2}\left( {{b^3} – {a^3}} \right)}
\end{array}} \right|$ અને ${\Delta _2} = \left| {\begin{array}{{20}{c}}
a&{{b^2}}&{{c^3}} \\
{{a^4}}&{{b^5}}&{{c^6}} \\
{{a^7}}&{{b^8}}&{{c^9}}
\end{array}} \right|$ તો ${\Delta _1}{\Delta _2}$ મેળવો.

normal

જો $\Delta = \left| {\,\begin{array}{{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ અને ${A_1},{B_1},{C_1}$ એ ઘટકો ${a_1},{b_1},{c_1}$ ના સહઅવયવ દર્શાવે છે તો નિશ્રાયક $\left| {\begin{array}{{20}{c}}{{A_1}}&{{B_1}}&{{C_1}}\\{{A_2}}&{{B_2}}&{{C_2}}\\{{A_3}}&{{B_3}}&{{C_3}}\end{array}} \right|$ નું મૂલ્ય મેળવો.

easy