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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q12E (page 39)

There exists a $2 脳 2$ matrix such that $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

Short Answer

Expert verified

True, there exist a $2 脳 2$ matrix A such thatrole="math" localid="1659642050851" $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

Step by step solution

01

Step 1:Taking the matrix

Suppose the matrix A is.

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

Now according to the given equation,

$A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

02

Justification of answer

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

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

After solving the equation we get

$[\begin{matrix} a + 2 b \\ c + 2 d \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

Now on equating the value of matrix we get

$a + 2 b = 3 c + 2 d = 4$

On solving the equations we get the value of variables

$a = 3 - 2 b c = 4 - 2 d$

Where, are free variables which can have any value. For instance let the value of $b, d = 1$

It implies $a = 1, c = 2$

Thus, we get the matrix

$A = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}]$

Hence, it is truethat there exist a $2 脳 2$ matrix A such that $A [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}]$

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Most popular questions from this chapter

Recall that a real square matrix A is called skew symmetric if $A^{T = - A}$ .

a. If A is skew symmetric, is $A^{2}$ skew symmetric as well? Or is $A^{2}$ symmetric?

b. If is skew symmetric, what can you say about the definiteness of $A^{2}$ ? What about the eigenvalues of $A^{2}$ ?

c. What can you say about the complex eigenvalues of a skew-symmetric matrix? Which skew-symmetric matrices are diagonalizable over $鈩�$ ?

Let be an orthogonal 2X2 matrix. Use the image of the unit circle to find the singular values of A.

The system is $\begin{matrix} [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 0 & 0 \end{matrix}] \end{matrix} \overset{鈫�}{x} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ inconsistent

In Exercises 57 through 61, consider a quadratic form q on $R^{3}$ with symmetric matrix A, with the given properties. In each case, describe the level surface $q (\overset{鈬赌}{x}) = 1$ geometrically.

59.qis positive semidefinite and rank A = 1.

Let A be a 4 脳 3 matrix, and let $\overset{鈫�}{b}$ and $\overset{鈫�}{c}$ be two vectors in ${鈩�}^{4}$ . We are told that the systemrole="math" localid="1659341825668" $A \overset{鈫�}{x} = \overset{鈫�}{b}$ has a unique solution. What can you say about the number of solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{c}$ ?

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