Question-3

Let \(A\) be a \(2\times2\) matrix, which is given as \(\begin{bmatrix}a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}\). Consider the following matrices:

\[ \begin{aligned} B & =\begin{bmatrix}a_{11}-a_{21} & a_{12}-a_{22}\\ a_{21} & a_{22} \end{bmatrix},C=\begin{bmatrix}a_{11}-a_{12} & a_{12}\\ a_{21}-a_{22} & a_{22} \end{bmatrix}\\ \\D & =\begin{bmatrix}a_{11}+a_{21} & a_{12}-a_{22}\\ a_{21} & a_{22} \end{bmatrix},E=\begin{bmatrix}a_{11}-a_{21} & a_{12}+a_{22}\\ a_{21} & a_{22} \end{bmatrix} \end{aligned} \]

Select all matrices that have the same determinant as that of matrix \(A\).

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• \(B\) is obtained by performing the operation \(R_{1}\rightarrow R_{1}-R_{2}\) on \(A\).

• \(C\) is obtained by performing the operation \(C_{1}\rightarrow C_{1}-C_{2}\) on \(A\).

We therefore have \(|B|=|C|=|A|\). There is no relationship between \(|D|\), \(|E|\) and \(|A|\).