Q44E For which values of constants a,... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q44E (page 346)

For which values of constants a,b,c are the matrix diagonalizable?
$[\begin{matrix} a & b \\ c & d \end{matrix}]$

Short Answer

Expert verified

The matrix A is diagonalizable for all values of $a, b, c 鈭� {鈩�}^{3}$

Step by step solution

Calculating the Characteristic Equation of the matrix .

Consider an $n 脳 n$ matrix A and a scalar $位$ .Then is the eigenvalue of A if $\det (A - 位濒) = 0$ . This is called Characteristic equation of matrix A.

For the given matrix $[\begin{matrix} a & b \\ c & d \end{matrix}]$ , we need to solve for,

$\det (A - 位濒) = 0 |\begin{matrix} a - 位 & b \\ b & c - 位 \end{matrix}| = 0 (a - 位) (c - 位) - b^{2} = 0 位^{2} - (a + c) 位 + (ac - b^{2}) = 0 位_{12} = \frac{a + c 卤 \sqrt{{(a + c)}^{2} - 4 ac - 4 b^{2}}}{2} = \frac{a + c 卤 \sqrt{{(a + c)}^{2} + 4 b^{2}}}{2}$

In the above expression, the expression under the square root is always greater than or equal to 0.

So If either $a 鈮� c, b 鈮� 0$ , we can have two distinct eigenvalues of a $2 脳 2$ matrix. So matrix A is diagonalizable.

Analyzing with different values for .

Even if the values a = c, b = 0, the eigenvalue is $位_{12} = a = c$ . Then we can solve for,

$(A - 位 l) x = 0 {[\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]}_{x_{2}}^{x_{1}} = 0 E_{伪} = {鈩�}^{2}$

Finally we can conclude that is diagonalizable for all $a, b, c 鈭� 鈩�$

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91影视

For which values of constants a,b,c are the matrix diagonalizable?
$[\begin{matrix} a & b \\ c & d \end{matrix}]$

Short Answer

Step by step solution

Calculating the Characteristic Equation of the matrix .

Analyzing with different values for .

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91影视

For which values of constants a,b,c are the matrix diagonalizable?[abcd]

Short Answer

Step by step solution

Calculating the Characteristic Equation of the matrix .

Analyzing with different values for .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

For which values of constants a,b,c are the matrix diagonalizable?
$[\begin{matrix} a & b \\ c & d \end{matrix}]$