
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
164.4k+ views
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.
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.
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NEET 2025 – Every New Update You Need to Know

Verb Forms Guide: V1, V2, V3, V4, V5 Explained

NEET Total Marks 2025

1 Billion in Rupees
