Q66 E 聽Find all lower triangular 3脳3... [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: Q66 E (page 86)

Find all lower triangular $3 脳 3$ matrices X such that $X^{3}$ is the zero matrix.

Short Answer

Expert verified

The matrix X such that $X^{3}$ is the zero matrix is $[\begin{matrix} 0 & 0 & 0 \\ q & 0 & 0 \\ r & s & 0 \end{matrix}]$ , where is any arbitrary.

Step by step solution

Assuming the matrix

Let the upper triangular matrix of $3 脳 3$ is $[\begin{matrix} 0 & 0 & 0 \\ q & 0 & 0 \\ r & s & 0 \end{matrix}]$

Now

$X = [\begin{matrix} a & 0 & 0 \\ b & d & 0 \\ c & e & f \end{matrix}] X^{2} = [\begin{matrix} a & 0 & 0 \\ b & d & 0 \\ c & e & f \end{matrix}] [\begin{matrix} a & 0 & 0 \\ b & d & 0 \\ c & e & f \end{matrix}] = [\begin{matrix} a^{2} & 0 & 0 \\ a b + b d & d^{2} & 0 \\ a c + e b + f c & e d + e f & f \end{matrix}] X^{3} = [\begin{matrix} a^{2} & 0 & 0 \\ a b + b d & d^{2} & 0 \\ a c + e b + f c & e d + e f & f \end{matrix}] [\begin{matrix} a & 0 & 0 \\ b & d & 0 \\ c & e & f \end{matrix}] = [\begin{matrix} a 3 & 0 & 0 \\ a (a b + b d) + d^{2 b} & d^{3} & 0 \\ a (a c + e b + f c) + b (e d + e f) + f c & d (e d + e f) & f^{3} \end{matrix}]$

Solving the Matrix

Since we have to put it equal to 0.

$X^{3} = [\begin{matrix} a^{2} & 0 & 0 \\ a b + b d & d^{2} & 0 \\ a c + e b + f c + b (e d + e f) & d (e d + e f) & f^{3} \end{matrix}] [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

On equating the matrix values,

$a^{3} = 0, d^{3} = 0, f^{3} = 0, a c + c b = 0 鈬� a = 0, b = 0 c (a + b) = 0$

Thus, the matrix X such that $X^{3}$ is the zero matrix is $[\begin{matrix} 0 & 0 & 0 \\ q & 0 & 0 \\ r & s & 0 \end{matrix}]$ .Where is any arbitrary.

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91影视

Find all lower triangular $3 脳 3$ matrices X such that $X^{3}$ is the zero matrix.

Short Answer

Step by step solution

Assuming the matrix

Solving the Matrix

One App. One Place for Learning.

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91影视

Find all lower triangular 3脳3matrices X such that X3is the zero matrix.

Short Answer

Step by step solution

Assuming the matrix

Solving the Matrix

One App. One Place for Learning.

Most popular questions from this chapter

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Find all lower triangular $3 脳 3$ matrices X such that $X^{3}$ is the zero matrix.