A real matrix A of size 2 × 2 has trace 9 and determinant 20. What is the trace of A²?

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Trace of a square

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A real matrix $A$ of size $2 \times 2$ has trace $9$ and determinant $20$ . What is the trace of $A^{2}$ ?

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Hint

Write the trace of $A^{2}$ in terms of the eigenvalues of $A$ .

Solution

If $A$ has eigenvalues $λ_{1}$ and $λ_{2}$ , then $λ_{1} + λ_{2} = 9$ and $λ_{1} λ_{2} = 20$ .
The eigenvalues of $A^{2}$ are $λ_{1}^{2}$ and $λ_{2}^{2}$ , so $tr (A^{2}) = λ_{1}^{2} + λ_{2}^{2}$ .
Then $λ_{1}^{2} + λ_{2}^{2} = (λ_{1} + λ_{2})^{2} - 2 λ_{1} λ_{2} = 81 - 40 = 41$ .

Answer: $41$