Question

The value of a for which the matrix \[\begin{array}{*{20}{c}}A& = &{\left[ {\begin{array}{*{20}{c}}a&2\\2&4\end{array}} \right]}\end{array}\] is singular if
A. \[\begin{array}{*{20}{c}}a& \ne &1\end{array}\]
B. \[\begin{array}{*{20}{c}}a& = &1\end{array}\]
C. \[\begin{array}{*{20}{c}}a& = &0\end{array}\]
D. \[\begin{array}{*{20}{c}}a& = &{ - 1}\end{array}\]

Answer 1

Hint: In this question, first of all, we will determine the determinant of matrix A. we know that if the matrix is a singular matrix, then the determinant of matrix A must be equal to Zero. After determining the determinant of a matrix, A, we will get a linear equation. Now, we will apply the condition of the singular matrix as we know it. After that, we will solve the linear equation. Hence, we will get a suitable answer.

Formula used:
If a matrix’s two rows are equal, then
 \[\begin{array}{*{20}{c}}{\left| A \right|}& = &0\end{array}\]

Complete step by step Solution:
In the question, we have given a square matrix A such as
\[\begin{array}{*{20}{c}}{ \Rightarrow A}& = &{\left[ {\begin{array}{*{20}{c}}a&2\\2&4\end{array}} \right]}\end{array}\]
Now, we will determine the determinant of matrix A. Therefore, we will get
\[\begin{array}{*{20}{c}}{ \Rightarrow \left| A \right|}& = &{\left| {\begin{array}{*{20}{c}}a&2\\2&4\end{array}} \right|}\end{array}\]
Now,
\[\begin{array}{*{20}{c}}{ \Rightarrow \left| A \right|}& = &{4a - 4}\end{array}\]
Now we have given the singular matrix. We know the condition if the matrix is singular, then the determinant of matrix A will be zero. Therefore, we can write it as,
\[\begin{array}{*{20}{c}}{ \Rightarrow \left| A \right|}& = &0\end{array}\]
Therefore,
\[\begin{array}{*{20}{c}}{ \Rightarrow 4a - 4}& = &0\end{array}\]
Now we will solve the above equation. Therefore, we will get the value of a,
\[\begin{array}{*{20}{c}}{ \Rightarrow a}& = &1\end{array}\].

Therefore, the correct option is (B).

Note: In this question, the first point is to keep in mind that if the matrix is a singular matrix, then the determinant of matrix A must be zero. A negative integer in the second matrix, for example, must be calculated carefully since it must be added rather than removed. Students mostly make mistakes in matrix problems that too in matrix multiplication problems. So, one should be cautious while solving these types of problems.