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Determinant of $$$\left[\begin{array}{cc}\frac{5}{2} - \lambda & \frac{3}{2}\\- \frac{3}{2} & - \lambda - \frac{1}{2}\end{array}\right]$$$
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Your Input
Calculate $$$\left|\begin{array}{cc}\frac{5}{2} - \lambda & \frac{3}{2}\\- \frac{3}{2} & - \lambda - \frac{1}{2}\end{array}\right|$$$.
Solution
The determinant of a 2x2 matrix is $$$\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$$$.
$$$\left|\begin{array}{cc}\frac{5}{2} - \lambda & \frac{3}{2}\\- \frac{3}{2} & - \lambda - \frac{1}{2}\end{array}\right| = \left(\frac{5}{2} - \lambda\right)\cdot \left(- \lambda - \frac{1}{2}\right) - \left(\frac{3}{2}\right)\cdot \left(- \frac{3}{2}\right) = \lambda^{2} - 2 \lambda + 1$$$
Answer
$$$\left|\begin{array}{cc}\frac{5}{2} - \lambda & \frac{3}{2}\\- \frac{3}{2} & - \lambda - \frac{1}{2}\end{array}\right| = \left(\lambda - 1\right)^{2}$$$A