Question

If \( A \) is a 3 \(\times\) 3 matrix with eigenvalues 1, 2, and 3, what is the trace of \( A^2 - 3A + I \)?

• A. -2
• B. 1
• C. 2
• D. -1

Hint:

For a matrix polynomial like \( A^2 - 3A + I \), apply the polynomial to the eigenvalues of the matrix to find the trace. The trace is the sum of the resulting eigenvalues.

Answer 1

The Correct Option is D

The trace of a matrix is the sum of its eigenvalues. Given that \( A \) is a \( 3 \times 3 \) matrix with eigenvalues 1, 2, and 3, we will use the fact that for any matrix \( A \), the trace of a polynomial of \( A \) is the same as the polynomial applied to the eigenvalues of \( A \).
First, let's calculate the trace of \( A^2 - 3A + I \). We can do this by applying the polynomial to the eigenvalues of \( A \), which are 1, 2, and 3.
For an eigenvalue \( \lambda \) of \( A \), the corresponding eigenvalue of \( A^2 - 3A + I \) is: \[ \lambda^2 - 3\lambda + 1 \] Now, we apply this to each eigenvalue:
- For \( \lambda = 1 \):
\( 1^2 - 3(1) + 1 = 1 - 3 + 1 = -1 \)
- For \( \lambda = 2 \):
\( 2^2 - 3(2) + 1 = 4 - 6 + 1 = -1 \)
- For \( \lambda = 3 \):
\( 3^2 - 3(3) + 1 = 9 - 9 + 1 = 1 \)
The trace is the sum of these values: \[ {Trace}(A^2 - 3A + I) = (-1) + (-1) + 1 = -1 \] Thus, the trace of \( A^2 - 3A + I \) is \( -1 \).