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question_answer1) The cofactor of the element '4' in the determinant \[\left| \,\begin{matrix} 1 & 3 & 5 & 1 \\ 2 & 3 & 4 & 2 \\ 8 & 0 & 1 & 1 \\ 0 & 2 & 1 & 1 \\ \end{matrix}\, \right|\] is [MP PET 1987]
question_answer2) If \[\Delta =\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix}\, \right|\] and \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\]denote the co-factors of \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\] respectively, then the value of the determinant \[\left| \begin{matrix} {{A}_{1}} & {{B}_{1}} & {{C}_{1}} \\ {{A}_{2}} & {{B}_{2}} & {{C}_{2}} \\ {{A}_{3}} & {{B}_{3}} & {{C}_{3}} \\ \end{matrix} \right|\] is [MP PET 1989]
question_answer3) If in the determinant \[\Delta =\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|\], \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\] etc. be the co-factors of \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\]etc., then which of the following relations is incorrect
question_answer4) If \[\omega \] is a cube root of unity and \[\Delta =\left| \begin{matrix} 1 & 2\omega \\ \omega & {{\omega }^{2}} \\ \end{matrix} \right|\], then \[{{\Delta }^{2}}\]is equal to [RPET 1984]
question_answer5) If \[{{\Delta }_{1}}=\left| \,\begin{matrix} 1 & 0 \\ a & b \\ \end{matrix}\, \right|\] and \[{{\Delta }_{2}}=\left| \begin{matrix} 1 & 0 \\ c & d \\ \end{matrix} \right|\], then \[{{\Delta }_{2}}{{\Delta }_{1}}\]is equal to [RPET 1984]
question_answer6) If \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\].... are respectively the co-factors of the elements \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\],...... of the determinant \[\Delta =\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix}\, \right|\], then \[\left| \begin{matrix} {{B}_{2}} & {{C}_{2}} \\ {{B}_{3}} & {{C}_{3}} \\ \end{matrix} \right|=\]
question_answer7) Let \[A={{[{{a}_{ij}}]}_{n\times n}}\]be a square matrix and let \[{{c}_{ij}}\]be cofactor of \[{{a}_{ij}}\]in A. If \[C=[{{c}_{ij}}]\],then
question_answer8) \[\left| \,\begin{matrix} {{\log }_{2}}512 & {{\log }_{4}}3 \\ {{\log }_{3}}8 & {{\log }_{4}}9 \\ \end{matrix}\, \right|\times \left| \,\begin{matrix} {{\log }_{2}}3 & {{\log }_{8}}3 \\ {{\log }_{3}}4 & {{\log }_{3}}4 \\ \end{matrix}\, \right|\]= [Tamilnadu (Engg.) 2002]
question_answer9) If \[A=\left| \,\begin{matrix} 5 & 6 & 3 \\ -4 & 3 & 2 \\ -4 & -7 & 3 \\ \end{matrix}\, \right|\,,\]then cofactors of the elements of 2nd row are [RPET 2002]
question_answer10) The minors of - 4 and 9 and the co-factors of - 4 and 9 in determinant \[\,\left| \,\begin{matrix} -1 & -2 & 3 \\ -4 & -5 & -6 \\ -7 & 8 & 9 \\ \end{matrix}\, \right|\] are respectively [J & K 2005]
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