Question

Suppose $a_1,a_2,...$ are real numbers, with $a_1\ne 0$. If $a_1,a_2,a_3,...$ are in A.P. Then,

• A. $A=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_5& a_6& a_7\end{array}\right]$ is singular
• B. the system of equations $a_{1}x+a_{2}y+a_{3}z=0,a_{4}x+a_{5}y+a_{6}z=0,a_{7}x+a_{8}y+a_{9}z=0$ has infinite number of solutions
• C. $B=\left[\begin{array}{cc}{a}_1& i{a}_2\\ i{a}_2& a_1\end{array}\right]$ is non singular
• D. none of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A $A=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_5& a_6& a_7\end{array}\right]$ is singular
B the system of equations $a_{1}x+a_{2}y+a_{3}z=0,a_{4}x+a_{5}y+a_{6}z=0,a_{7}x+a_{8}y+a_{9}z=0$ has infinite number of solutions
C $B=\left[\begin{array}{cc}{a}_1& i{a}_2\\ i{a}_2& a_1\end{array}\right]$ is non singular

Given $a_1,a_2...$are in AP

$a_1\ne 0$

Option A$=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right]$

$=\left[\begin{array}{ccc}{a}_1& a_1+d& a_1+2d\\ a_1+3d& a_1+4d& a_1+5d\\ a_1+4d& a_1+5d& a_1+6d\end{array}\right]C_1\to C_2-C_2$

$=\left[\begin{array}{ccc}d& a_1+d& a_1+2d\\ d& a_1+4d& a_1+5d\\ d& a_1+5d& a_1+6d\end{array}\right]C_2\to C_3-C_2$

$=\left[\begin{array}{ccc}d& d& a_1+2d\\ d& d& a_1+5d\\ d& d& a_1+6d\end{array}\right]$

$C_1\&C_2$ are same $\Rightarrow △=0$

The matrix is singular-correct

Option B - co efficients matrix$=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right]$

They are in AP

$\Rightarrow △=0$

$\therefore$They have infinite solutions

Option B is also correct

Option C $B=\left[\begin{array}{cc}{a}_1& i{a}_2\\ i{a}_2& a_1\end{array}\right]$

$\Rightarrow △=a_1^2-\left(i^2{a}_2^2\right)$

$=a_1^2+a_2^2$

$=a_1^2+{(a_1+d)}^2$

$=2a_1^2+d^2+2a_{2}d>0$

Non singular

Option A,B, C are true