Question

If A = [a_ij] is a square matrix of even order such that a_ij = i² − j², then
(a) A is a skew-symmetric matrix and | A | = 0
(b) A is symmetric matrix and | A | is a square
(c) A is symmetric matrix and | A | = 0
(d) none of these.

Answer 1

(d) none of these


$$\begin{gathered} \mathrm{Given}:A\mathrm{is}\mathrm{a}\mathrm{square}\mathrm{matrix}\mathrm{of}\mathrm{even}\mathrm{order}. \\ \mathrm{Let}A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right] \\ \Rightarrow A=\left[\begin{array}{cc}0& -3\\ 3& 0\end{array}\right]\left[\because a_{ij}=i^2-j^2\right] \\ \mathrm{So},\mathrm{it}\mathrm{is}\mathrm{a}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}\mathrm{as}{a}_{ij}=-a_{ji}. \\ \mathrm{Now}, \\ \left|A\right|=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=\left[\begin{array}{c}{a}_{11}{a}_{22}-a_{21}{a}_{12}\end{array}\right]=\left[\begin{array}{c}0-\left(-9\right)\end{array}\right]=9 \end{gathered}$$