Question

Consider the following statements:
1. The matrix
$\left(\begin{array}{ccc}1& 2& 1\\ a& 2a& 1\\ b& 2b& 1\end{array}\right)$ is singular.
2. The matrix
$\left(\begin{array}{ccc}c& 2c& 1\\ a& 2a& 1\\ b& 2b& 1\end{array}\right)$ is non-singular.
Which of the above statements is/are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is A 1 only

For matrix

$\left|\begin{array}{ccc}1& 2& 1\\ a& 2a& 1\\ b& 2b& 1\end{array}\right|$

$C_2\to C_2-2C_1$

$\left|\begin{array}{ccc}1& 0& 1\\ a& 0& 1\\ b& 0& 1\end{array}\right|$ The determinant is $0$. So the matrix is singular.

For matrix,

$\left|\begin{array}{ccc}c& 2c& 1\\ a& 2a& 1\\ b& 2b& 1\end{array}\right|$

$C_2\to C_2-2C_1$

$\left|\begin{array}{ccc}c& 0& 1\\ a& 0& 1\\ b& 0& 1\end{array}\right|$ The determinant is $0$. So the matrix is singular.

Thus, only $1$ is true.

Hence, option A is correct.