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[Solution Library] Let M=(lla b , c d). Show: det; M=1/2(tr; M)^2-1/2 tr;(M^2) Suppose M is a 3 * 3 matrix. Find and verify a similar formula for det; M

Question: Let \(M=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)\). Show:

\[\operatorname{det} M=\frac{1}{2}(\operatorname{tr} M)^{2}-\frac{1}{2} \operatorname{tr}\left(M^{2}\right)\]

Suppose \(M\) is a \(3 \times 3\) matrix. Find and verify a similar formula for \(\operatorname{det} M\) in terms of \(\operatorname{tr}\left(M^{3}\right), \operatorname{tr}\left(M^{2}\right)\), and tr \(M\). Hint: make an ansatz for your formula and derive a system of linear equations for any unknowns you introduce by testing it on explicit matrices.

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