Chapter 1: Q16 (page 39)

Question: Determine whether the statements that follow are true or false, and justify your answer.
16: There exists a 2x2 matrix such that
$A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}] a n d A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$ .

Short Answer

Expert verified

Answer:

False, there doesn’t exist a 2x2 matrix A such that $A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}] a n d A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$ .

Step by step solution

Taking the matrix

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

Now according to the given equations

$A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}] a n d A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$ .

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

$[\begin{matrix} 21 + 2 b \\ 2 c + 2 d \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$

Now on equating the value of matrix we get

a + b = 1

2c + 2d = 1

On solving the equations, we get the value of variables

Since, the above equations can’t be solved.

Hence, there doesn’t exist a 2x2 matrix A such that $A [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}] a n d A [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$ .

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Short Answer

Step by step solution

Taking the matrix

Justification of answer

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Question: Determine whether the statements that follow are true or false, and justify your answer.16: There exists a 2x2 matrix such thatA[11]=[12]andA[22]=[21].

Short Answer

Step by step solution

Taking the matrix

Justification of answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Pure Maths

Discrete Mathematics

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.