Use Sage to do your computations for you.
See Appendix B.5: Triangular block form.
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Carry out steps a, b, c below for each of the following matrices.
- \(A = \begin{bmatrix} -5 & 3 & 1 \\ -4 & 2 & 1 \\ -4 & 3 & 0 \end{bmatrix}\)
- \(A = \begin{bmatrix} -3 & 6 & 3 & 2 \\ -2 & 3 & 2 & 2 \\ -1 & 3 & 0 & 1 \\ -1 & 1 & 2 & 0 \end{bmatrix}\)
NOTE:
These are the same matrices as in the second problem from the practise problems for scalar-triangular form,
and again from the practise problems for triangular block form.
If you have not already done so, you should complete those problems first.
- Compute the characteristic polynomial \(c_A(\lambda)\).
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Verify the Cayley-Hamilton Theorem for \(A\).
That is, verify that \(c_A(A)\) is the zero matrix.
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Form the companion matrix of the polynomial \( p(x) = x^4 + x^3 + 4 x^2 - 5 x - 2 \),
and then compute the characteristic polynomial of that matrix.
What do you notice?
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Verify that the characteristic polynomial of a companion matrix is always equal to its corresponding monic, degree-\(n\) polynomial.
(See Companion matrices in the textbook.)
Answers