Question

If \[\left[ {\begin{array}{*{20}{c}}
  {2 + x}&3&4 \\
  1&{ - 1}&2 \\
  x&1&{ - 5}
\end{array}} \right]\]is a singular matrix, then x is
A. \[\dfrac{{13}}{{25}}\]
B. \[ - \dfrac{{25}}{{13}}\]
C. \[\dfrac{5}{{13}}\]
D. \[\dfrac{{25}}{{13}}\]

Answer 1

Hint: A singular matrix refers to a matrix whose determinant is zero. To get the required value of x, calculate the determinant of the given 3x3 matrix and equate it to 0. From the equation thus obtained, the value of x can be found.

Formula used:
The determinant of a \[3 \times 3\] matrix \[A = \left[ {\begin{array}{*{20}{c}}
  a&b&c \\
  d&e&f \\
  g&h&i
\end{array}} \right]\] is \[\left| A \right| = a \times \left| {\begin{array}{*{20}{c}}
  e&f \\
  h&i
\end{array}} \right| - b \times \left| {\begin{array}{*{20}{c}}
  d&f \\
  g&i
\end{array}} \right| + c \times \left| {\begin{array}{*{20}{c}}
  d&e \\
  g&h
\end{array}} \right|\]

Complete step by step solution:
A singular matrix refers to a matrix whose determinant is zero.
Let \[A = \left[ {\begin{array}{*{20}{c}}
  {2 + x}&3&4 \\
  1&{ - 1}&2 \\
  x&1&{ - 5}
\end{array}} \right]\]
\[\left| A \right| = \left| {\begin{array}{*{20}{c}}
  {2 + x}&3&4 \\
  1&{ - 1}&2 \\
  x&1&{ - 5}
\end{array}} \right| = \left( {2 + x} \right)\left( {5 - 2} \right) - 3( - 5 - 2x) + 4(1 + x) = 0\]
\[ \Rightarrow 6 + 3x + 15 + 6x + 4 + 4x = 0\]
\[ \Rightarrow 13x = - 25\]
\[ \Rightarrow x = - \dfrac{{25}}{{13}}\]

Option B. is the correct answer.

Note: To solve the given problem, one must know what a singular matrix is and its properties. One must also know to calculate the determinant of a 3x3 matrix. The determinant of a matrix can be found only if the matrix is a square matrix.