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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 2: Q55 E (page 86)

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 55.
$[\begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix}] X = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

Short Answer

Expert verified

The matrix X that will satisfy the equation will be $[\begin{matrix} - 2 b & - 2 d \\ b & d \end{matrix}]$

Step by step solution

01

Assuming the matrix

Since, the product of matrix gives $2 脳 2$ matrix then the order of matrix X should be $2 脳 2$ .

Let the matrix be $X = [\begin{matrix} a & c \\ b & d \end{matrix}]$ .

Then the equation becomes,

$[\begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix}] [\begin{matrix} a & c \\ b & d \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

02

Solving matrix

For solving the given matrix equation first multiply the left-hand side matrix.

$[\begin{matrix} a + 2 b & c + 2 d \\ 2 a + 4 b & 2 c + 4 c \end{matrix}] [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

Now equating the left-hand and right-hand matrix.

$a + 2 b = 0 c + 2 d = 0 2 a + 4 b = 0 2 c + 4 d = 0$

On solving the above equations, we will get

$a = - 2 b, b = b, c = - 2 d, d = d$

Hence, the matrix X that will satisfy the equation will be $[\begin{matrix} - 2 b & - 2 d \\ b & d \end{matrix}]$

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Most popular questions from this chapter

If A is a $3 脳 4$ matrix and B is a $4 脳 5$ , then AB will be a $5 脳 3$ matrix.

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises $16$ through $23$ . Show the effect of these transformations on the letter $L$ considered in Example $5$ . In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise $13$ .

20. $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$

Consider a linear transformation Tfrom $R^{2}$ to $R^{2}$ . Suppose that $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two arbitrary vectors in $R^{2}$ and that $\overset{鈫�}{x}$ is a third vector whose endpoint is on the line segment connecting the endpoints of $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ . Is the endpoint of the vector $T (\overset{鈫�}{x)}$ necessarily on the line segment connecting the endpoints of $T (\overset{鈫�}{v)}$ and $T (\overset{鈫�}{w)}$ ? Justify your answer.

Find the matrices of the transformations Tand Ldefined in Exercise 80.

There exists a positive integer n such that ${[\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]}^{n} = l_{2}$ .

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