Q16E Determine whether the statements... [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 1: Q16E (page 39)

Determine whether the statements that follow are true or false, and justify your answer.
16: There exists a $2 脳 2$ matrix such that
$A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$

Short Answer

Expert verified

False, there doesn鈥檛 exist a matrix A such that $A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$ .

Step by step solution

Step 1:Taking the matrix

Suppose the matrix A is. $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$

Now according to the given equations

role="math" localid="1659673249605" $A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$

Justification of answer

Put the value of A matrix in the above equation we get.

$[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$

After solving the equation we get

$[\begin{matrix} a + b \\ c + d \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$

Now on equating the value of matrix we get

$a + b = 1 c + d = 1$

We also have another equation.

$[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$

After solving the equation we get

$[\begin{matrix} 2 a + 2 b \\ 2 c + 2 d \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$

Now on equating the value of matrix we get

$a + b = 1 2 c + 2 d = 1$

On solving the equations, we get the value of variables

Since, the above equations can鈥檛 be solved.

Hence, there doesn鈥檛 exist a $2 脳 2$ matrix A such that $A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$

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91影视

Determine whether the statements that follow are true or false, and justify your answer.
16: There exists a $2 脳 2$ matrix such that
$A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$

Short Answer

Step by step solution

Step 1:Taking the matrix

Justification of answer

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91影视

Determine whether the statements that follow are true or false, and justify your answer.16: There exists a 2脳2matrix such thatA[11]=[12]andA[22]=[21]

Short Answer

Step by step solution

Step 1:Taking the matrix

Justification of answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Determine whether the statements that follow are true or false, and justify your answer.
16: There exists a $2 脳 2$ matrix such that
$A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}]$ and $A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$