If $p(t)$ is the characteristic polynomial for an $n\times n$ matrix $A$, then the matrix $p(A)$ is the $n \times n$ zero matrix.
Solution.
We use the Cayley-Hamilton theorem.
The Cayley-Hamilton Theorem.
If $p(t)$ is the characteristic polynomial for an $n\times n$ matrix $A$, then the matrix $p(A)$ is the $n \times n$ zero matrix.
To obtain the characteristic polynomial for $T$, we note that the matrix $T$ is upper triangular. Thus $T-tI$ is also upper triangular and recall that the determinant of an upper triangular matrix is the product of the diagonal entries. Thus the characteristic polynomial $p_T(t)$ for $T$ is
\[p_T(t)=\det(T-tI)=(1-t)(1-t)(2-t)=-t^3+4t^2-5t+2.\]
By the Cayley-Hamilton theorem, we have
\[p_T(T)=-T^3+4T^2-5T+2I=O.\]
(Don’t forget $I$.)
Here $O$ is the $3 \times 3$ zero matrix.
This problem is actually one of the final exam problems in the linear algebra class I taught.
Of course, you may calculate them directly but most students who computed directly made calculation mistakes (unfortunately).
The second common mistake is that students jumped to conclusion that the expression $-T^3+4T^2+5T-2I$ is the zero matrix after finding the characteristic polynomial.
Those students knew that they could use the Cayley-Hamilton theorem but were a bit careless. The given expression and the characteristic polynomial are slightly different.
Anyway, I didn’t subtract many points for the second mistake.
(And you get little points if you made a mistake for a direct calculation.)
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7 & a & b \\
0 &2 &c \\
0 & 0 & 3
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diagonalizable?
(The Ohio State University, Linear Algebra Final Exam Problem)
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Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.
What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?
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Let
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
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Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
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Let
\[A=\begin{bmatrix}
1 & -1\\
2& 3
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Hint.
Apply the Cayley-Hamilton theorem.
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