Q7-39E If A is a 2脳2聽matrix such that... [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: Q7-39E (page 383)

If A is a $2 脳 2$ matrix such that $t r A = 1$ androle="math" localid="1668513563951" $d e t A = - 6$ ,then A must be diagonalizable.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Find the determinant of a matrix

$\begin{array}{rcl} A & = & - 6 \\ a (1 - a) - b c & = & - 6 \\ a - a^{2} - b c & = & - 6 \end{array}$ Here,

$A = [\begin{matrix} a & b \\ c & 1 - a \end{matrix}]$

Then we have,

$\begin{array}{rcl} A & = & - 6 \\ a (1 - a) - b c & = & - 6 \\ a - a^{2} - b c & = & - 6 \end{array}$

Explanation for the statement

Now, solve for the characteristic equation:

$\begin{array}{rcl} \det (A - 位 I) & = & 0 \\ |\begin{matrix} a - 位 & b \\ c & 1 - a - 位 \end{matrix}| & = & 0 \\ - a^{2} - a 位 - 位 + a 位 + 位^{2} - b c & = & 0 \\ 位^{2} - 位 + (a (1 - a) - b c) & = & 0 \\ 位^{2} - 位 - 6 & = & 0 \\ 位 & = & 3, - 2 \end{array}$

As trace is 1, diagonal elements are $a, 1 - a$ . It is non that the trace of any matrix is the sum of the main diagonal elements of the matrix. Then we have

$\begin{array}{rcl} a + (1 - a) & = & 1 \\ 1 & = & 1 \end{array}$

Which is true, so the elements of the main diagonal can be the eigenvalues and rest of the elements can be 0. Then the matrix becomes,

$A = [\begin{matrix} 3 & 0 \\ 0 & - 2 \end{matrix}]$

Hence, the matrix A is diagonalizable.

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91影视

If A is a $2 脳 2$ matrix such that $t r A = 1$ androle="math" localid="1668513563951" $d e t A = - 6$ ,then A must be diagonalizable.

Short Answer

Step by step solution

Find the determinant of a matrix

Explanation for the statement

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91影视

If A is a 2脳2matrix such thattrA=1androle="math" localid="1668513563951" detA=-6,then A must be diagonalizable.

Short Answer

Step by step solution

Find the determinant of a matrix

Explanation for the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Decision Maths

Geometry

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

If A is a $2 脳 2$ matrix such that $t r A = 1$ androle="math" localid="1668513563951" $d e t A = - 6$ ,then A must be diagonalizable.