12th NCERT Matrices Exercise 4.4 Questions 5
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Write Minors and Cofactors of the elements of following dterminants

Question (1)

\[(i)\left| {\begin{array}{*{20}{c}}2&{ - 4}\\0&3\end{array}} \right|\;\;\;\;\left( {ii} \right)\left| {\begin{array}{*{20}{c}}a&c\\b&d\end{array}} \right|\]

Solution

\[\left( i \right)\left| {\begin{array}{*{20}{c}}2&{ - 4}\\0&3\end{array}} \right|\] \[{M_{11}} = 3\quad {M_{21}} = - 4\] \[{M_{12}} = 0\quad {M_{22}} = 2\] \[MinA = \left| {\begin{array}{*{20}{c}}3&0\\{ - 4}&2\end{array}} \right|\] \[CofactorA = \left| {\begin{array}{*{20}{c}}3&0\\4&2\end{array}} \right|\]
\[\left( {ii} \right)\left| {\begin{array}{*{20}{c}}a&c\\b&d\end{array}} \right|\] \[MinA = \left| {\begin{array}{*{20}{c}}d&b\\c&a\end{array}} \right|\] \[CofactorA = \left| {\begin{array}{*{20}{c}}d&{ - b}\\{ - c}&a\end{array}} \right|\]

Question (2)

\[(i)\left| {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right|\;\;\;\;\left( {ii} \right)\left| {\begin{array}{*{20}{c}}1&0&4\\3&5&{ - 1}\\0&1&2\end{array}} \right|\]

Solution

\[(i)\left| {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right|\] \[{M_{11}} = \left| {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right| = 1\] \[{M_{12}} = \left| {\begin{array}{*{20}{c}}0&0\\0&1\end{array}} \right| = 0\] \[MinA = \left| {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right|\] \[CofactorA = \left| {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right|\]
\[\left( {ii} \right)\left| {\begin{array}{*{20}{c}}1&0&4\\3&5&{ - 1}\\0&1&2\end{array}} \right|\] \[MinA = \left| {\begin{array}{*{20}{c}}{11}&6&3\\{ - 4}&2&1\\{ - 20}&{ - 13}&5\end{array}} \right|\] \[CofactorA = \left| {\begin{array}{*{20}{c}}{11}&{ - 6}&3\\4&2&{ - 1}\\{ - 20}&{13}&5\end{array}} \right|\]

Question (3)

Using Cofactors of elements of second row, evalutae
\[\Delta = \left| {\begin{array}{*{20}{c}}5&3&8\\2&0&1\\1&2&3\end{array}} \right|\;\]

Solution

\[\Delta = \left| {\begin{array}{*{20}{c}}5&3&8\\2&0&1\\1&2&3\end{array}} \right|\;\] \[{a_{21}} = 2,{a_{22}} = 0,{a_{23}} = 1\] \[{A_{21}} = {\left( { - 1} \right)^{2 + 1}}\left| {\begin{array}{*{20}{c}}3&8\\2&3\end{array}} \right|\;\] \[{A_{21}} = - 1\left( {9 - 16} \right) = 7\] \[{A_{22}} = {\left( { - 1} \right)^{2 + 2}}\left( {15 - 8} \right) = 7\] \[{A_{23}} = {\left( { - 1} \right)^{2 + 3}}\left( {10 - 3} \right) = - 7\] \[\Delta = {a_{21}}{A_{21}} + {a_{22}}{A_{22}} + {a_{23}}{A_{23}}\] \[\Delta = 2\left( 7 \right) + 0\left( 7 \right) + 1\left( { - 7} \right)\] \[\Delta = 14 + 0 - 7 = 7\]

Question (4)

Using Cofactors of elements of third column, evalutae
\[\Delta = \left| {\begin{array}{*{20}{c}}1&x&{yz}\\1&y&{zx}\\1&z&{xy}\end{array}} \right|\]

Solution

\[\Delta = \left| {\begin{array}{*{20}{c}}1&x&{yz}\\1&y&{zx}\\1&z&{xy}\end{array}} \right|\] \[{a_{13}} = yz;{a_{23}} = zx;{a_{33}} = xy\] \[{A_{13}} = - \left( {z - y} \right)\] \[{A_{13}} = - \left( {z - y} \right),{A_{23}} = - \left( {z - x} \right),{A_{33}} = \left( {y - z} \right)\] \[\Delta = {a_{13}}{A_{13}} + {a_{23}}{A_{23}} + {a_{33}}{A_{33}}\] \[\Delta = yz\left( {z - y} \right) + \left[ { - zx\left( {z - x} \right)} \right] + xy\left( {y - x} \right)\] \[\Delta = y{z^2} - {y^2}z - {z^2}x + {x^2}z + x{y^2} - {x^2}y\] \[\Delta = - {x^2}y + {x^2}z + x{y^2} - {z^2}x - {y^2}z + y{z^2}\] \[\Delta = - {x^2}\left( {y - z} \right) + x\left( {{y^2} - {z^2}} \right) - yz\left( {y - z} \right)\] \[\Delta = \left( {y - z} \right)\left[ { - {x^2} + x\left( {y + z} \right) - yz} \right]\] \[\Delta = \left( {y - z} \right)\left[ { - {x^2} + xy + xz - yz} \right]\] \[\Delta = \left( {y - z} \right)\left[ { - x\left( {x - y} \right) + z\left( {x - y} \right)} \right]\] \[\Delta = \left( {y - z} \right)\left( {x - y} \right)\left( { - x + z} \right)\] \[\Delta = \left( {x - y} \right)\left( {y - z} \right)\left( {z - x} \right)\]

Question (5)

\[If\;\Delta = \left| {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right|\] and Aij is Cofactors of aij, then value of δ is given by
\[\left( A \right){a_{11}}{A_{31}} + {a_{12}}{A_{32}} + {a_{13}}{A_{33}}\] \[\left( B \right){a_{11}}{A_{11}} + {a_{12}}{A_{21}} + {a_{13}}{A_{31}}\] \[\left( C \right){a_{21}}{A_{11}} + {a_{22}}{A_{12}} + {a_{23}}{A_{13}}\] \[\left( D \right){a_{11}}{A_{11}} + {a_{21}}{A_{21}} + {a_{31}}{A_{31}}\]

Solution

\[\Delta = \left| {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right|,{A_{ij}} =\text{cofactor of}\;{a_{ij}}\] \[\Delta = {a_{11}}{A_{11}} + {a_{21}}{A_{21}} + {a_{31}}{A_{31}}\] So Option (D) is correct
Exercise 4.3⇐
⇒Exercise 4.5