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 (where $i=\sqrt{-1}$)
• 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 nonsingular
• 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 (where $i=\sqrt{-1}$)
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 nonsingular

$\left(A\right)$ $A=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_5& a_6& a_7\end{array}\right]=\left[\begin{array}{ccc}d& d& a_3\\ d& d& a_6\\ d& d& a_7\end{array}\right]=0C_1\to C_2-C_1,C_2\to C_3-C_2{C}_1=C_2$

where $d$ is the common ratio

$\left(B\right)$ $a_{1}x+a_{2}y+a_{3}z=0a_{4}x+a_{5}y+a_{6}z=0a_{7}x+a_{8}y+a_{9}z=0$

For infinite solution, coefficient matrix $=0$

$\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right]=\left[\begin{array}{ccc}d& d& a_3\\ d& d& a_6\\ d& d& a_9\end{array}\right]=0C_1\to C_1-C_2,C_2\to C_3-C_2{C}_1=C_2$

$\left(C\right)$ $B=\left[\begin{array}{cc}{a}_1& i{a}_2\\ i{a}_2& a_1\end{array}\right]=({a_1}^2+{a_2}^2)\ne 0$