How to Calculate and Simplify a Matrix Polynomial

Ohio State University exam problems and solutions in mathematics

Problem 47

Let $T=\begin{bmatrix}
1 & 0 & 2 \\
0 &1 &1 \\
0 & 0 & 2
Calculate and simplify the expression
\[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix.

(The Ohio State University Linear Algebra Exam)

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Use 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.


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

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.

Now we compute
&=p_T(T)+10T-4I=10T-4I\\[6pt] &=\begin{bmatrix}
10 & 0 & 20 \\
0 &10 &10 \\
0 & 0 & 20

4 & 0 & 0 \\
0 &4 &0 \\
0 & 0 & 4
\end{bmatrix}\\[6pt] &=\begin{bmatrix}
6 & 0 & 20 \\
0 &6 &10 \\
0 & 0 & 16
Therefore the answer is
6 & 0 & 20 \\
0 &6 &10 \\
0 & 0 & 16



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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