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Question
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
Options
A is a skew-symmetric matrix and | A | = 0
A is symmetric matrix and | A | is a square
A is symmetric matrix and | A | = 0
none of these.
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Solution
none of these
\[\text{Given: A is a square matrix of even order} . \]
\[\]
\[Let A = \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}\]
\[ \Rightarrow A = \begin{bmatrix}0 & - 3 \\ 3 & 0\end{bmatrix} \left[ \because a_{ij} = i^2 - j^2 \right]\]
\[\]
\[\text{So, it is a skew - symmetric matrix as a_{ij} }= - a_{ji} . \]
\[Now, \]
\[\left| A \right| = \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} = \begin{bmatrix}a_{11} a_{22} - a_{21} a_{12}\end{bmatrix} = \begin{bmatrix}0 - \left( - 9 \right)\end{bmatrix} = 9\]
\[\]
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