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Chapter 2: Problem 33

Let $$ A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -6 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 4 & 8 \\ -7 & 3 \end{array}\right] $$ a. Find \(A^{T}\) and show that \(\left(A^{T}\right)^{T}=A\). b. Show that \((A+B)^{T}=A^{T}+B^{T}\). c. Show that \((A B)^{T}=B^{T} A^{T}\).

Short Answer

Expert verified
In summary, we get the following results: a. \(A^T = \begin{bmatrix} 2 & 5 \\ 4 & -6 \end{bmatrix}\), and \((A^T)^T = A\). b. \((A+B)^T = \begin{bmatrix} 6 & -2 \\ 12 & -3 \end{bmatrix}\), which is equal to \(A^T + B^T\). c. (AB)^T = \(\begin{bmatrix} -16 & 58 \\ 32 & 50 \end{bmatrix}\), which is equal to \(B^T A^T\).

Step by step solution

01

Finding A transpose (A^T)

To find the transpose of matrix A, interchange its rows and columns: $$ A^T = \left[\begin{array}{cc} 2 & 5 \\ 4 & -6 \end{array}\right] $$
02

Verifying (A^T)^T = A

Now, take the transpose of matrix A^T by interchanging its rows and columns: $$ \left(A^T\right)^T = \left[\begin{array}{cc} 2 & 4 \\ 5 & -6 \end{array}\right] $$ As we can see, \((A^T)^T\) = A.
03

Finding B transpose (B^T)

Find the transpose of matrix B, interchange its rows and columns: $$ B^T = \left[\begin{array}{cc} 4 & -7 \\ 8 & 3 \end{array}\right] $$
04

Showing (A+B)^T = A^T + B^T

First, find A+B: $$ A+B = \left[\begin{array}{cc} 2 & 4 \\ 5 & -6 \end{array}\right] + \left[\begin{array}{cc} 4 & 8 \\ -7 & 3 \end{array}\right] = \left[\begin{array}{cc} 6 & 12 \\ -2 & -3 \end{array}\right] $$ Now, find the transpose of A+B: $$ (A+B)^T = \left[\begin{array}{cc} 6 & -2 \\ 12 & -3 \end{array}\right] $$ And finally, find A^T + B^T: $$ A^T + B^T = \left[\begin{array}{cc} 2 & 5 \\ 4 & -6 \end{array}\right] + \left[\begin{array}{cc} 4 & -7 \\ 8 & 3 \end{array}\right] = \left[\begin{array}{cc} 6 & -2 \\ 12 & -3 \end{array}\right] $$ As we can see, (A+B)^T = A^T + B^T
05

Showing (AB)^T = B^T A^T

First, find the product of A and B: $$ AB = \left[\begin{array}{cc} 2 & 4 \\ 5 & -6 \end{array}\right] \cdot \left[\begin{array}{cc} 4 & 8 \\ -7 & 3 \end{array}\right] = \left[\begin{array}{cc} -16 & 32 \\ 58 & 50 \end{array}\right] $$ Now, find the transpose of the product AB: $$ (AB)^T = \left[\begin{array}{cc} -16 & 58 \\ 32 & 50 \end{array}\right] $$ And finally, find the product of B^T and A^T: $$ B^T A^T = \left[\begin{array}{cc} 4 & -7 \\ 8 & 3 \end{array}\right] \cdot \left[\begin{array}{cc} 2 & 5 \\ 4 & -6 \end{array}\right] = \left[\begin{array}{cc} -16 & 58 \\ 32 & 50 \end{array}\right] $$ As we can see, (AB)^T = B^T A^T.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Addition
Matrix addition is a fundamental operation in linear algebra. It involves adding corresponding elements from two matrices of the same dimensions. If matrices don鈥檛 have the same dimensions, they cannot be directly added. This operation is simple and retains the same dimensions as the initial matrices.

For example, consider two matrices:
  • Matrix A: \[A = \begin{bmatrix} 2 & 4 \ 5 & -6 \end{bmatrix}\]
  • Matrix B: \[B = \begin{bmatrix} 4 & 8 \ -7 & 3 \end{bmatrix}\]
To add these matrices, simply add each corresponding element:
\[A + B = \begin{bmatrix} 2+4 & 4+8 \ 5+(-7) & -6+3 \end{bmatrix} = \begin{bmatrix} 6 & 12 \ -2 & -3 \end{bmatrix}\]

This straightforward operation is useful for solving systems of linear equations and other applications within linear algebra.
Matrix Multiplication
Matrix multiplication is slightly more complex than matrix addition. It involves taking the dot products of rows and columns. However, they must follow specific rules for dimensions: the number of columns in the first matrix must be equal to the number of rows in the second matrix.

Continuing with matrices A and B as mentioned earlier, the product of these matrices is:
  • Matrix A (2x2) and matrix B (2x2) can be multiplied because the column count of A matches the row count of B.
Compute the multiplication as follows:\[AB = \begin{bmatrix} 2 & 4 \ 5 & -6 \end{bmatrix} \cdot \begin{bmatrix} 4 & 8 \ -7 & 3 \end{bmatrix} = \begin{bmatrix} (2*4 + 4*(-7)) & (2*8 + 4*3) \ (5*4 + (-6)*7) & (5*8 + (-6)*3) \end{bmatrix} = \begin{bmatrix} -16 & 32 \ 58 & 50 \end{bmatrix}\]

This product allows for complex transformations of data through linear mappings.
Linear Algebra
Linear algebra is a branch of mathematics focussing on vectors, matrices, and linear transformations. It is applicable in various fields including computer science, engineering, and natural sciences. At its core, linear algebra deals with linear equations and their representations through matrices and vector spaces.

Key concepts include:
  • Vectors: One-dimensional arrays representing magnitude and direction.
  • Matrices: Two-dimensional arrays that can operate on vectors and other matrices.
  • Matrix Operations: Addition, multiplication, transposition, and inversion.
  • Vector Spaces: A mathematical structure formed by a collection of vectors.
These elements are crucial for understanding systems of linear equations and transformation properties such as rotations, scaling, and more. Linear algebra simplifies complex problems, making it an essential tool in various technical and scientific applications.

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Most popular questions from this chapter

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. A dietitian wishes to plan a meal around three foods. The percent of the daily requirements of proteins, carbohydrates, and iron contained in each ounce of the three foods is summarized in the following table: $$\begin{array}{lccc} \hline & \text { Food I } & \text { Food II } & \text { Food III } \\ \hline \text { Proteins }(\%) & 10 & 6 & 8 \\ \hline \text { Carbohydrates }(\%) & 10 & 12 & 6 \\ \hline \text { Iron }(\%) & 5 & 4 & 12 \\ \hline \end{array}$$ Determine how many ounces of each food the dietitian should include in the meal to meet exactly the daily requirement of proteins, carbohydrates, and iron \((100 \%\) of each).

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{aligned} x_{1}-2 x_{2}+x_{3} &=6 \\ 2 x_{1}+x_{2}-3 x_{3} &=-3 \\ x_{1}-3 x_{2}+3 x_{3} &=10 \end{aligned} $$

Let $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] $$ a. Find \(A^{-1}\) if it exists. b. Find a necessary condition for \(A\) to be nonsingular. c. Verify that \(A A^{-1}=A^{-1} A=I\).

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. The management of Hartman Rent-A-Car has allocated $$\$ 1.5$$ million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full- size cars. Compacts cost $$\$ 12,000$$ each, intermediatesize cars cost $$\$ 18,000$$ each, and full-size cars cost $$\$ 24,000$$ each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100 , determine how many cars of each type will be purchased. (Assume that the entire budget will be used.)

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{l} 2 x+3 y-2 z=10 \\ 3 x-2 y+2 z=0 \\ 4 x-y+3 z=-1 \end{array} $$
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