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If $x, y$ are any two non-zero real numbers, $a_{i j}=x i+y j, A=\left\{a_{i j}\right\}_{n \times n}$ and $P, Q$ are two $n \times n$ matrices such that $A=x P+y Q$, then
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Verified Answer
The correct answer is:
$P+Q$ is symmetric and $P-Q$ is skew symmetric
We have, $a_{i j}=x j+y i$,
$$
\begin{aligned}
& A=\left(a_i\right)_{n \times n} \\
& A=\left[\begin{array}{cccc}
x+y & 2 x+y & 3 x+y & \ldots . . n x+y \\
x+2 y & 2 x+2 y & 3 x+2 y & \ldots . . n x+2 y \\
x+3 y & \ldots & \ldots . . & \ldots . \\
x+n y & \ldots & \ldots . & n x+n y
\end{array}\right]_{n x n n} \\
& A=x\left[\begin{array}{cccc}
1 & 2 & 3 \ldots . . & n \\
1 & 2 & 3 \ldots . & n \\
1 & \vdots & \ldots . . & \\
1 & 2 & 3 \ldots . & n
\end{array}\right]+y\left[\begin{array}{cccc}
1 & 1 & 1 . \ldots . & 1 \\
2 & \ldots & \ldots & 2 \\
3 & \ldots . & \ldots & 3 \\
\ldots & \ldots & \ldots . & n
\end{array}\right] \\
& A=x P+y Q \\
&
\end{aligned}
$$
where,
$$
P=\left[\begin{array}{cccc}
1 & 2 & 3 \ldots \ldots & n \\
1 & 2 & 3 \ldots & n \\
1 & 2 & \ldots . & \ldots \\
1 & 2 & 3 \ldots & n
\end{array}\right] \text { and } Q=\left[\begin{array}{cccc}
1 & 1 & 1 \ldots . . & 1 \\
2 & 2 & 2 \ldots . & 2 \\
3 & \ldots & \ldots & 3 \\
\ldots n & \ldots & n \ldots & n
\end{array}\right]
$$
$\therefore(P+Q)$ is symmetric and $(P-Q)$ is skew symmetric.
$$
\begin{aligned}
& A=\left(a_i\right)_{n \times n} \\
& A=\left[\begin{array}{cccc}
x+y & 2 x+y & 3 x+y & \ldots . . n x+y \\
x+2 y & 2 x+2 y & 3 x+2 y & \ldots . . n x+2 y \\
x+3 y & \ldots & \ldots . . & \ldots . \\
x+n y & \ldots & \ldots . & n x+n y
\end{array}\right]_{n x n n} \\
& A=x\left[\begin{array}{cccc}
1 & 2 & 3 \ldots . . & n \\
1 & 2 & 3 \ldots . & n \\
1 & \vdots & \ldots . . & \\
1 & 2 & 3 \ldots . & n
\end{array}\right]+y\left[\begin{array}{cccc}
1 & 1 & 1 . \ldots . & 1 \\
2 & \ldots & \ldots & 2 \\
3 & \ldots . & \ldots & 3 \\
\ldots & \ldots & \ldots . & n
\end{array}\right] \\
& A=x P+y Q \\
&
\end{aligned}
$$
where,
$$
P=\left[\begin{array}{cccc}
1 & 2 & 3 \ldots \ldots & n \\
1 & 2 & 3 \ldots & n \\
1 & 2 & \ldots . & \ldots \\
1 & 2 & 3 \ldots & n
\end{array}\right] \text { and } Q=\left[\begin{array}{cccc}
1 & 1 & 1 \ldots . . & 1 \\
2 & 2 & 2 \ldots . & 2 \\
3 & \ldots & \ldots & 3 \\
\ldots n & \ldots & n \ldots & n
\end{array}\right]
$$
$\therefore(P+Q)$ is symmetric and $(P-Q)$ is skew symmetric.
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