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3 and 4 .Determinants and Matrices
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Suppose $a_1, a_2, .......$ real numbers, with $a_1 \ne 0$. If $a_1, a_2, a_3, ..........$ are in $A.P$. then
A$A =$ $\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{a_4}}&{{a_5}}&{{a_6}}\\{{a_5}}&{{a_6}}&{{a_7}}\end{array}} \right]$ is singular
Bthe system of equations $a_1x + a_2y + a_3z = 0, a_4x + a_5y + a_6z = 0, a_7x + a_8y + a_9z = 0$ has infinite number of solutions
C$B =$$\left[ {\begin{array}{*{20}{c}}{{a_1}}&{i{a_2}}\\{i{a_2}}&{{a_1}}
\end{array}} \right]$ is non singular ; where $i =$$\,\sqrt { - 1} \,$
DAll of the above
Solution
Let We have $|A| =$ $\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{a_4}}&{{a_5}}&{{a_6}}\\{{a_5}}&{{a_6}}&{{a_7}}\end{array}} \right]$ $=$ $\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\
{3d}&{3d}&{3d}\\d&d&d\end{array}} \right]$ $=$ $0$
[ Using $R_3 \rightarrow R_3 -R_2,$ and $R_2 \rightarrow R_2 -R_1$ ]
==> $A$ is singular
$\therefore$ The given system of homogeneous equations has infinite number of solutions.
Also $|B| =$ $a_1^2 + a_2^2 \ne 0$. Thus $B$ is non- singular
Standard 12
Mathematics
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