The value of determinant$\left( {\begin{array}{*{20}{c}}

  {19}&6&7 \\

  {21}&3&{15} \\

  {28}&{11}&6

\end{array}} \right)$is:

$(a)$150

$(b)$-110

$(c)$0

$(d)$None of these

Hint: In the above given question, we are asked to find the determinant of the given matrix. The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the given matrix.

Let us assume the given matrix to be:

A$ = \left( {\begin{array}{*{20}{c}}

  {19}&6&7 \\

  {21}&3&{15} \\

  {28}&{11}&6

\end{array}} \right)$

Now, we know that the determinant of a matrix ${\text{A}}$ is given as$\left| {\text{A}} \right|$.

If${\text{A = }}\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)$

then, \[\left| {\text{A}} \right|{\text{ = }}{a_{11}}\left( {\begin{array}{*{20}{c}}

  {{a_{22}}}&{{a_{23}}} \\

  {{a_{32}}}&{{a_{33}}}

\end{array}} \right) - {a_{12}}\left( {\begin{array}{*{20}{c}}

  {{a_{21}}}&{{a_{23}}} \\

  {{a_{31}}}&{{a_{33}}}

\end{array}} \right) + {a_{13}}\left( {\begin{array}{*{20}{c}}

  {{a_{21}}}&{{a_{22}}} \\

  {{a_{31}}}&{{a_{32}}}

\end{array}} \right)\] … (1)

Therefore, we using the equation (1), we get

 \[\left| {\text{A}} \right|{\text{ = 19}}\left( {\begin{array}{*{20}{c}}

  3&{15} \\

  {11}&6

\end{array}} \right) - 6\left( {\begin{array}{*{20}{c}}

  {21}&{15} \\

  {28}&6

\end{array}} \right) + 7\left( {\begin{array}{*{20}{c}}

  {21}&3 \\

  {28}&{11}

\end{array}} \right)\]

$ = 19(3 \times 6 - 11 \times 15) - 6(21 \times 6 - 15 \times 28) + 7(21 \times 11 - 3 \times

28)$

$ = 19(18 - 165) - 6(126 - 420) + 7(231 - 84)$

$ = 19( - 147) - 6( - 294) + 7(147)$

$ = - 2739 + 1764 + 1029$

$ = - 2739 + 2739$

$ = 0$

Hence, the determinant of the given matrix is option$(c)$ 0.

Note: When we face such a type of problem, the key point is to have a good understanding of the matrices and their properties. Then, put the values from the given matrix in the formula for calculating the determinant directly and further evaluate it for obtaining the correct solution.