Question-10

If the diagonal entries of a \(3\times3\) lower triangular matrix \(A\) are \(1,2,3\), find the sum of the roots of the equation \(\text{det}(A-xI)=0\), where \(I\) is the identity matrix.

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\(A-xI\) is also lower triangular. Since the determinant of a lower triangular matrix is the product of the diagonal entries, we have:

\[ \begin{aligned} \text{det}(A-xI) & =(1-x)(2-x)(3-x) \end{aligned} \]

The roots of \(\text{det}(A-xI)=0\) are \(1,2,3\). So the sum of the roots is equal to \(6\).