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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: Q57 E (page 86)

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation.
$[\begin{matrix} 1 & 2 \\ 3 & 5 \end{matrix}] X = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Short Answer

Expert verified

The matrix X that will satisfy the equation will be $[\begin{matrix} - 5 & 2 \\ 3 & 1 \end{matrix}]$

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 will become

$[\begin{matrix} 1 & 2 \\ 3 & 5 \end{matrix}] [\begin{matrix} a & c \\ b & d \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

02

Solving the Matrix

To solve the given matrix equation, first multiply the left-hand side matrix.

$[\begin{matrix} a + 2 b & c + 2 d \\ 3 a + 5 b & 3 c + 5 d \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

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

$a + 2 b = 1 c + 2 d = 0 3 a + 5 b = 0 3 c + 5 d = 1$

On solving the above equations, we will get

$a = - 5, b = 3, c = 2, d = - 1$

Hence, the matrix which satisfied the given equation is $[\begin{matrix} - 5 & 2 \\ 3 & - 1 \end{matrix}]$

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Most popular questions from this chapter

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation $T \overset{鈫�}{(x)} = A \overset{鈫�}{x}$ on this face.

28. $[\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}]$

Some parking meters in downtown Geneva, Switzerland, accept $2$ Franc and $5$ Franc coins.

a. A parking officer collects $51$ coins worth $144$ Francs. How many coins are there of each kind?

b. Find the matrix $A$ that transforms the vector

$[\begin{matrix} number of 2 Franc coins \\ number of5Franc coins \end{matrix}]$

into the vector

$[\begin{matrix} total value of coins \\ totalnumberof coins \end{matrix}]$

c. Is the matrix $A$ in part (b) invertible? If so, find the inverse (use Exercise 13). Use the result to check your answer in part (a).

Which of the functions f from R toR in Exercises 21 through 24 are invertible?22 $f (x) = 2^{x}$ .

Consider a block matrix $A = [\begin{matrix} A_{11} & 0 \\ 0 & A_{22}^{} \end{matrix}]$ , where role="math" localid="1660371822794" $A_{11} and A_{12}$ are square matrices. For which choices of $A_{11} and A_{22}$ is A invertible? In these cases, what is ?

If matrices A and B commute, then the formula A²B = BA² must hold.

See all solutions

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