Q29E For which values of the constant... [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 2: Q29E (page 97)

For which values of the constant k is the following matrix invertible?
$[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & k \\ 1 & 4 & k^{2} \end{matrix}]$

Short Answer

Expert verified

For each kdistinct than 1 or 2.

Step by step solution

Check the given matrix is invertible or not

The given matrix will be invertible if and only if its reduced row echelon form is $l_{3}$ . Therefore, let transform the given matrix by using the row operation

$[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & k \\ 1 & 4 & k^{2} \end{matrix}] R_{2} 鈫� R_{2} - R_{1}, R_{3} 鈫� R_{3} - R_{1} [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & k - 1 \\ 0 & 3 & k^{2} - 1 \end{matrix}] R_{3} 鈫� R_{3} - 3 R_{z} [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & k - 1 \\ 0 & 0 & k^{2} - 3 k + 2 \end{matrix}]$

It is clear know that reduced row echelon form of the given matrix will be except for solutions of the equations

$k^{2} - 3 k + 2 = 0$

Find the value of k

Because in that case the entries of third row are all .Therefore, rewriting the left side of the equation as,

$k^{2} - 3 k + 2 = k^{2} - 2 k - k + 2 = k (k - 2) - (k - 2) = (k - 2) (k - 1)$

We obtain that its solution is role="math" localid="1660369236997" $k = 1 and k = 2$ .

Hence, the given matrix is invertible for each kexcept $k = 1 and k = 2$ .

For each distinct than or 2.

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91影视

For which values of the constant k is the following matrix invertible?
$[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & k \\ 1 & 4 & k^{2} \end{matrix}]$

Short Answer

Step by step solution

Check the given matrix is invertible or not

Find the value of k

One App. One Place for Learning.

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91影视

For which values of the constant k is the following matrix invertible?[11112k14k2]

Short Answer

Step by step solution

Check the given matrix is invertible or not

Find the value of k

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Decision Maths

Discrete Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

For which values of the constant k is the following matrix invertible?
$[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & k \\ 1 & 4 & k^{2} \end{matrix}]$