A matrix A of size 3 × 3 has eigenvalues 1, 2 and 4. Compute det(A² − 2A + 3I).

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Polynomial of a matrix

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A matrix $A$ of size $3 \times 3$ has eigenvalues $1$ , $2$ and $4$ . Compute $det (A^{2} - 2 A + 3 I)$ .

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Hint

A polynomial in $A$ acts on each eigenvector of $A$ as the same polynomial in the eigenvalue.

Solution

Let $p (t) = t^{2} - 2 t + 3$ ; then $p (A)$ has eigenvalues $p (1)$ , $p (2)$ and $p (4)$ .
These are $p (1) = 1 - 2 + 3 = 2$ , $p (2) = 4 - 4 + 3 = 3$ and $p (4) = 16 - 8 + 3 = 11$ .
The determinant is the product of the eigenvalues, $2 \times 3 \times 11 = 66$ .

Answer: $66$