Matrix Equation Analysis
Given the equation \((A - 3I)(A - 5I) = O\), we can infer the eigenvalues of matrix \(A\). The factors imply that the eigenvalues of \(A\) are \(3\) and \(5\). Since \(A\) is a \(3 \times 3\) non-singular matrix, it has three eigenvalues, and the third eigenvalue must be non-zero.
Eigenvalues of A
- The eigenvalues of \(A\) must be \(3\), \(5\), and another eigenvalue \(\lambda\) such that \(\lambda \neq 0\).
- Since \(A\) is non-singular, all eigenvalues must be non-zero.
Determining the Product ab
Given that \(aA + bA^{-1} = 4I\), we can multiply both sides by \(A\):
\[
aA^2 + bI = 4A
\]
Rearranging gives:
\[
aA^2 - 4A + bI = 0
\]
This indicates that \(A\) satisfies the quadratic equation in terms of \(A\).
Using the eigenvalues, we substitute \(A\) with its eigenvalues in the equation:
- For \(\lambda = 3\):
\[
a(3^2) - 4(3) + b = 0 \implies 9a - 12 + b = 0 \quad (1)
\]
- For \(\lambda = 5\):
\[
a(5^2) - 4(5) + b = 0 \implies 25a - 20 + b = 0 \quad (2)
\]
Solving the System
From equations (1) and (2):
1. \(b = 12 - 9a\) from (1)
2. Substitute \(b\) in (2):
\[
25a - 20 + 12 - 9a = 0 \implies 16a - 8 = 0 \implies a = \frac{1}{2}
\]
Substituting \(a\) back to find \(b\):
\[
b = 12 - 9\left(\frac{1}{2}\right) = 12 - \frac{9}{2} = \frac{24 - 9}{2} = \frac{15}{2}
\]
Final Calculation
Now calculating \(ab\):
\[
ab = \left(\frac{1}{2}\right) \left(\frac{15}{2}\right) = \frac{15}{4}
\]
Thus, the product \(a b\) is \(\frac{15}{4}\).