Write Minors and Cofactors of elements of following determinants: left|begin{array}{lll}1 & 0 &a

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3 and 4 .Determinants and Matrices

easy

Write Minors and Cofactors of the elements of following determinants: $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$

Option A

Option B

Option C

Option D

Solution

The given determinant is $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$

By the definition of minors and cofactors, we have:

$\mathrm{M}_{11}=$ minor of $a_{11}=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|=1$

$\mathrm{M}_{12}=$ minor of $a_{12}=\left|\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right|=0$

$\mathrm{M}_{13}=$ minor of $a_{13}=\left|\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right|=0$

$\mathrm{M}_{21}=$ minor of $a_{21}=\left|\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right|=0$

$\mathrm{M}_{22}=$ minor of $a_{22}=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|=1$

$\mathrm{M}_{23}=$ minor of $a_{23}=\left|\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right|=0$

$\mathrm{M}_{31}=$ minor of $a_{31}=\left|\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right|=0$

$\mathrm{M}_{32}=$ minor of $a_{32}=\left|\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right|=0$

$\mathrm{M}_{33}=$ minor of $a_{33}=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|=1$

$\mathrm{A}_{11}=$ cofactor of $a_{11}=(-1)^{1+1} \mathrm{M}_{11}=1$

$\mathrm{A}_{12}=$ cofactor of $a_{12}=(-1)^{1+2} \mathrm{M}_{12}=0$

$\mathrm{A}_{13}=$ cofactor of $a_{13}=(-1)^{1+3} \mathrm{M}_{13}=0$

$\mathrm{A}_{21}=$ cofactor of $a_{21}=(-1)^{2+1} \mathrm{M}_{21}=0$

$\mathrm{A}_{22}=$ cofactor of $a_{22}=(-1)^{2+2} \mathrm{M}_{22}=1$

$\mathrm{A}_{23}=$ cofactor of $a_{23}=(-1)^{2+3} \mathrm{M}_{23}=0$

$\mathrm{A}_{31}=$ cofactor of $a_{31}=(-1)^{3+1} \mathrm{M}_{31}=0$

$\mathrm{A}_{32}=$ cofactor of $a_{32}=(-1)^{3+2} \mathrm{M}_{32}=0$

$\mathrm{A}_{33}=$ cofactor of $a_{33}=(-1)^{3+3} \mathrm{M}_{33}=1$

Standard 12

Mathematics

Write Minors and Cofactors of the elements of following determinants: $\left|\begin{array}{ll}a & c \\ b & d\end{array}\right|$

easy

If in the determinant $\Delta = \left| {\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right|$, ${A_1},{B_1},{C_1}$ $etc$. be the co-factors of ${a_1},{b_1},{c_1}$etc., then which of the following relations is incorrect

easy

$\left| {\,\begin{array}{{20}{c}}{{{\log }_3}512}&{{{\log }_4}3}\\{{{\log }_3}8}&{{{\log }_4}9}\end{array}\,} \right| \times \left| {\,\begin{array}{{20}{c}}{{{\log }_2}3}&{{{\log }_8}3}\\{{{\log }_3}4}&{{{\log }_3}4}\end{array}\,} \right|=$

medium

If $\Delta=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$ and $\mathrm{A}_{i j}$ is Cofactors of $a_{i j},$ then value of $\Delta$ is given by

easy

If $P = \left[ {\begin{array}{*{20}{c}}1&\alpha &3\\1&3&3\\2&4&4\end{array}} \right]$ is the adjoint of $3×3 $ matrix $A$ and $ |A|=4$ then $\alpha $ is equal to

medium

(JEE MAIN-2013)

Write Minors and Cofactors of the elements of following determinants: $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$

Solution

Similar Questions

Write Minors and Cofactors of the elements of following determinants: $\left|\begin{array}{ll}a & c \\ b & d\end{array}\right|$

If in the determinant $\Delta = \left| {\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right|$, ${A_1},{B_1},{C_1}$ $etc$. be the co-factors of ${a_1},{b_1},{c_1}$etc., then which of the following relations is incorrect

$\left| {\,\begin{array}{*{20}{c}}{{{\log }_3}512}&{{{\log }_4}3}\\{{{\log }_3}8}&{{{\log }_4}9}\end{array}\,} \right| \times \left| {\,\begin{array}{*{20}{c}}{{{\log }_2}3}&{{{\log }_8}3}\\{{{\log }_3}4}&{{{\log }_3}4}\end{array}\,} \right|=$

If $\Delta=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$ and $\mathrm{A}_{i j}$ is Cofactors of $a_{i j},$ then value of $\Delta$ is given by

If $P = \left[ {\begin{array}{*{20}{c}}1&\alpha &3\\1&3&3\\2&4&4\end{array}} \right]$ is the adjoint of $3×3 $ matrix $A$ and $ |A|=4$ then $\alpha $ is equal to

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$\left| {\,\begin{array}{{20}{c}}{{{\log }_3}512}&{{{\log }_4}3}\\{{{\log }_3}8}&{{{\log }_4}9}\end{array}\,} \right| \times \left| {\,\begin{array}{{20}{c}}{{{\log }_2}3}&{{{\log }_8}3}\\{{{\log }_3}4}&{{{\log }_3}4}\end{array}\,} \right|=$