Short Answer

Consider the matrix $A = | \begin{matrix} a & b \\ b & c \end{matrix} |$ 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 $a \neq c, b \neq 0$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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,

$d e t (A - λ l) = 0 d e t |\begin{matrix} a - λ & b \\ b & c - λ \end{matrix}| = 0 (a - λ) (c - λ) - b^{2} = 0 λ^{2} - (a + c) (c - λ) - b^{2} = 0 λ_{1, 2} = \frac{a + c \pm \sqrt{a^{2} + 2 a c + c^{2} - 4 a c + b^{2}}}{2} = \frac{a + c \pm \sqrt{{(a - c)}^{2} + b^{2}}}{2}$

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

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Short Answer

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

Step by Step Solution

Step 1: Eigenvalues

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

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Step by Step Solution

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

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