Question

$\Huge{\textmid{Solve\:it}}$

$\begin{array}{|ccc|}\sf 3 &\sf 6 &\sf 9 \\ \sf 4 &\sf 8 &\sf 12 \\ \sf 7 &\sf 14 &\sf 21\end{array}$
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Answer 1

$\large\underline\purple{\bold{Solution :- }}$

$\rm :\implies\:\begin{array}{|ccc|}\sf 3 &\sf 6 &\sf 9 \\ \sf 4 &\sf 8 &\sf 12 \\ \sf 7 &\sf 14 &\sf 21\end{array}$

$\rm :\:Taking \: 3 \: common \: from \: R_1, 4 \: from \: R_2, \: 7 \: from \: R_3$

$\rm :\implies\:3 \times 4 \times 7 \times \begin{array}{|ccc|}\sf 1 &\sf 2 &\sf 3 \\ \sf 1 &\sf 2 &\sf 3 \\ \sf 1 &\sf 2 &\sf 3\end{array}$

$\rm :\implies\:3 \times 4 \times 7 \times 0$

{If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.}

$\rm :\implies\:0$

$\rm :\implies\: \large\boxed{ \red{ \bf \: \begin{array}{|ccc|}\sf 3 &\sf 6 &\sf 9 \\ \sf 4 &\sf 8 &\sf 12 \\ \sf 7 &\sf 14 &\sf 21\end{array}= 0}}$

Additional Information -

• The determinant remains unaltered if its rows are changed into columns and the columns into rows. This is known as the property of reflection.

• If all the elements of a row (or column) are zero, then the determinant is zero.

• If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.

• The interchange of any two rows (or columns) of the determinant changes its sign.

• If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.