Let {Delta ₁} = left| {,begin{array}{*{20}{c}}{{a₁}}&{{b₁}}&{{c₁}}{{a₂}}&{{b?

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3 and 4 .Determinants and Matrices

medium

Let ${\Delta _1} = \left| {\,\begin{array}{{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ and ${\Delta _2} = \left| {\,\begin{array}{{20}{c}}{{\alpha _1}}&{{\beta _1}}&{{\gamma _1}}\\{{\alpha _2}}&{{\beta _2}}&{{\gamma _2}}\\{{\alpha _3}}&{{\beta _3}}&{{\gamma _3}}\end{array}\,} \right|$, then ${\Delta _1} \times {\Delta _2}$ can be expressed as the sum of how many determinants

$9$

$3$

$27$

$2$

Solution

(c) Each term in ${\Delta _1} \times {\Delta _2}$ is the sum of three terms. So each entry in ${C_1}$ or ${C_2}$ or ${C_3}$ in ${\Delta _1} \times {\Delta _2}$ is the sum of three terms. Hence, ${\Delta _1} \times {\Delta _2}$ can be written as the sum of $3 × 3 × 3 = 27$ determinants.

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 ${\Delta _1} = \left| {\begin{array}{{20}{c}}
{{b^5}{c^6}\left( {{c^3} – {b^3}} \right)}&{{a^4}{c^6}\left( {{a^3} – {c^3}} \right)}&{{a^4}{b^5}\left( {{b^3} – {a^3}} \right)} \\
{{b^2}{c^3}\left( {{b^6} – {c^6}} \right)}&{a{c^3}\left( {{c^6} – {a^6}} \right)}&{a{b^2}\left( {{a^6} – {b^6}} \right)} \\
{{b^2}{c^3}\left( {{c^3} – {b^3}} \right)}&{a{c^3}\left( {{a^3} – {c^3}} \right)}&{a{b^2}\left( {{b^3} – {a^3}} \right)}
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{{20}{c}}
a&{{b^2}}&{{c^3}} \\
{{a^4}}&{{b^5}}&{{c^6}} \\
{{a^7}}&{{b^8}}&{{c^9}}
\end{array}} \right|$ then ${\Delta _1}{\Delta _2}$ is equal to

normal

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 = \left| {\,\begin{array}{*{20}{c}}5&6&3\\{ – 4}&3&2\\{ – 4}&{ – 7}&3\end{array}\,} \right|\,, $ then cofactors of the elements of $2^{nd}$ row are

easy

The minors of $ -4$ and $ 9$ and the co-factors of $-4$ and $ 9$ in determinant $\,\left| {\,\begin{array}{*{20}{c}}{ – 1}&{ – 2}&3\\{ – 4}&{ – 5}&{ – 6}\\{ – 7}&8&9\end{array}\,} \right|$ are respectively

easy

Solution

Similar Questions

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

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 = \left| {\,\begin{array}{*{20}{c}}5&6&3\\{ – 4}&3&2\\{ – 4}&{ – 7}&3\end{array}\,} \right|\,, $ then cofactors of the elements of $2^{nd}$ row are

The minors of $ -4$ and $ 9$ and the co-factors of $-4$ and $ 9$ in determinant $\,\left| {\,\begin{array}{*{20}{c}}{ – 1}&{ – 2}&3\\{ – 4}&{ – 5}&{ – 6}\\{ – 7}&8&9\end{array}\,} \right|$ are respectively

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