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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: Q1E (page 1)

If possible, compute the matrix products in Exercises 1 through 13, using paper and pencil.
1. $[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Short Answer

Expert verified

Product of given matrix is. $[\begin{matrix} 4 & 6 \\ 3 & 4 \end{matrix}]$

Step by step solution

01

Step1:Matrix multiplication

If A is matrix of order $n 脳 p$ and B is matrix of order $m 脳 q$ . Then the matrix multiplication of $A B$ Is defined only if. $p = m$

If Bis a $m 脳 q$ matrix and A $n 脳 p$ matrix, then the product BAis defined as the matrix of the linear transformation. $T (\overset{鈫�}{x}) = B (A \overset{鈫�}{x})$

.

02

Assuming the matrix

Let the given matrix

$A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Order of Matrix $A$ is $2 脳 2$ and order of matrix $B$ is. $2 脳 2$

Since. $2 = 2$ Thus, the product is defined

03

Multiplication of matrix

Now, find the product as follows:

$A B = [\begin{matrix} 1.1 + 1.3 & 1.2 + 1.4 \\ 0.1 + 1.3 & 0.2 + 1.4 \end{matrix}] = [\begin{matrix} 4 & 6 \\ 3 & 4 \end{matrix}]$

Hence, the product of the given matrix is. $[\begin{matrix} 4 & 6 \\ 3 & 4 \end{matrix}]$

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Most popular questions from this chapter

Solve the linear system of equations. You may use technology.

$| \begin{matrix} 3 x_1 + 6 x_2 + 9 x_3 + 5 x_4 + 25 x_5 & = 53 \\ 7 x_1 + 14 x_2 + 21 x_3 + 9 x_4 + 53 x_5 & = 105 \\ - 4 x_1 - 8 x_2 - 12 x_3 + 5 x_4 - 10 x_5 & = 11 \end{matrix} |$

In Exercises 1 through 12, find all solutions of the equations

with paper and pencil using Gauss鈥揓ordan elimination.

Show all your work.

$| \begin{matrix} x_{4} + 2 x_{5} - x_{6} = 2 \\ x_{1} + 2 x_{2} + x_{5} - x_{6} = 0 \\ x_{1} + 2 x_{2} + 2 x_{3 -} x_{5} - x_{6} = 2 \end{matrix} |$

Determine whether the statements that follow are true or false, and justify your answer.

19. There exits a matrix A such that $A [\begin{matrix} - 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 5 \\ 7 \end{matrix}]$

in exercises 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions.

3. $|2 x + 4 y = 3 3 x + 6 y = 2|$

Use the concept of a continuous dynamical system.Solve the differential equation $\frac{d x}{d t} = 鈭� k x$ . Solvethe system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ whenAis diagonalizable overR,and sketch the phase portrait for 2脳2 matricesA.

Solve the initial value problems posed in Exercises 1through 5. Graph the solution.

4 . $\frac{d y}{d t} = 0.8 t$ with $y (0) = - 0.8$

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