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

In the determinant $\left| {\,\begin{array}{*{20}{c}}0&1&{ – 2}\\{ – 1}&0&3\\2&{ – 3}&0\end{array}\,} \right|$, the ratio of the co-factor to its minor of the element $-3$ is

easy

If $a_i^2 + b_i^2 + c_i^2 = 1,\,\,(i = 1,2,3)$ and ${a_i}{a_j} + {b_i}{b_j} + {c_i}{c_j} = 0$ $(i \ne j,i,j = 1,2,3)$ then the value of ${\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}\,} \right|^2}$ is

hard

Using Cofactors of elements of third column, evaluate $\Delta=\left|\begin{array}{ccc}1 & x & y z \\ 1 & y & z x \\ 1 & z & x y\end{array}\right|$

easy

Write Minors and Cofactors of the elements of following determinants : $\left|\begin{array}{rr}2 & -4 \\ 0 & 3\end{array}\right|$

easy

Find minors and cofactors of the elements of the determinant

$\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ and verify that $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}=0$

medium

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

In the determinant $\left| {\,\begin{array}{*{20}{c}}0&1&{ – 2}\\{ – 1}&0&3\\2&{ – 3}&0\end{array}\,} \right|$, the ratio of the co-factor to its minor of the element $-3$ is

If $a_i^2 + b_i^2 + c_i^2 = 1,\,\,(i = 1,2,3)$ and ${a_i}{a_j} + {b_i}{b_j} + {c_i}{c_j} = 0$ $(i \ne j,i,j = 1,2,3)$ then the value of ${\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}\,} \right|^2}$ is

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Find minors and cofactors of the elements of the determinant

$\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ and verify that $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}=0$