Q21E 21. Find a聽matrix A for which聽... [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: Q21E (page 345)

21. Find amatrix A for which $E_{1} = span [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $E_{2} = span [\begin{matrix} 2 \\ 3 \end{matrix}]$
How many such matrices are there?

Short Answer

Expert verified

The given matrix contains only one $A = (\begin{matrix} 5 & - 2 \\ 6 & - 2 \end{matrix})$

Step by step solution

Algebraic Versus.

Algebraic versus geometric multiplicity If 位 is an eigenvalues of a square matrix A,

then $gemu (1) < almu (1)$

role="math" localid="1659593735913" $A = (\begin{matrix} a & b \\ c & d \end{matrix}) Let E_{1} = span ([\begin{matrix} 1 \\ 2 \end{matrix}]) A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}] We have (\begin{matrix} a & b \\ c & d \end{matrix}) [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}] a + 2 b = 1, c + 2 d = 2 a = 1 - 2 b, c = 2 - 2 d$

Now we need to put for second value.

$E_{2} = s p a n ([\begin{matrix} 2 \\ 3 \end{matrix}]) A [\begin{matrix} 2 \\ 3 \end{matrix}] = [\begin{matrix} 4 \\ 6 \end{matrix}]$

$(\begin{matrix} a & b \\ c & d \end{matrix}) [\begin{matrix} 2 \\ 3 \end{matrix}] = [\begin{matrix} 4 \\ 6 \end{matrix}] (\begin{matrix} 1 - 2 b & b \\ 2 - 2 b & d \end{matrix}) [\begin{matrix} 2 \\ 3 \end{matrix}] = [\begin{matrix} 4 \\ 6 \end{matrix}]$

$2 (1 - 2 b) + 3 b = 4, 2 (2 - 2 d) + 3 d = 6 b = - 2, d = - 2, a = 5, c = 6$

Therefore, There is only one such matrix $A = (\begin{matrix} 5 & - 2 \\ 6 & - 2 \end{matrix})$

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21. Find amatrix A for which $E_{1} = span [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $E_{2} = span [\begin{matrix} 2 \\ 3 \end{matrix}]$
How many such matrices are there?

Short Answer

Step by step solution

Algebraic Versus.

Now we need to put for second value.

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21. Find amatrix A for whichE1=span[12] andE2=span[23]How many such matrices are there?

Short Answer

Step by step solution

Algebraic Versus.

Now we need to put for second value.

One App. One Place for Learning.

Most popular questions from this chapter

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Discrete Mathematics

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21. Find amatrix A for which $E_{1} = span [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $E_{2} = span [\begin{matrix} 2 \\ 3 \end{matrix}]$
How many such matrices are there?