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Q16E

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Found in: Page 336

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Consider the matrix ${\mathbit{A}}{\mathbf{=}}|\begin{array}{cc}a& b\\ b& c\end{array}|$ where a, b, and c are nonzero constants. For which values of a, b, and c does A have two distinct eigenvalues?

Values of a, b, and c for which A have two distinct eigenvalues $a\ne c,b\ne 0$

See the step by step solution

## Step 1: Eigenvalues

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.

## Step 2: Finding values of a, b, and c for which A have two distinct eigenvalues:

We can clearly see that,

$det\left(A-\lambda l\right)=0\phantom{\rule{0ex}{0ex}}det\left|\begin{array}{ccc}a-\lambda & & b\\ b& & c-\lambda \end{array}\right|=0\phantom{\rule{0ex}{0ex}}\left(a-\lambda \right)\left(c-\lambda \right)-{b}^{2}=0\phantom{\rule{0ex}{0ex}}{\lambda }^{2}-\left(a+c\right)\left(c-\lambda \right)-{b}^{2}=0\phantom{\rule{0ex}{0ex}}{\lambda }_{1,2}=\frac{a+c±\sqrt{{a}^{2}+2ac+{c}^{2}-4ac+{b}^{2}}}{2}\phantom{\rule{0ex}{0ex}}=\frac{a+c±\sqrt{{\left(a-c\right)}^{2}+{b}^{2}}}{2}$

Hence, $a\ne c,b\ne 0$.