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Chapter 3: Problem 27

$$ \left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right] $$

Short Answer

Expert verified
The determinant of the given matrix is -2, its inverse is \(\left[\begin{array}{cc} 1/2 & 1/2 \\ 1/2 & -1/2 \end{array}\right]\), and its transpose is \(\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]\).

Step by step solution

01

Calculate the determinant

To calculate the determinant of a 2x2 matrix, use the following formula: \(\text{det}(A) = ad - bc\), where elements of matrix A are represented as: \[ A = \left[\begin{array}{cc} a & b \\ c & d \end{array}\right] \] For the given matrix: \[ \left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right] \] we have a = 1, b = 1, c = 1, and d = -1. Thus, \(\text{det}(A) = (1 脳 -1) - (1 脳 1)\).
02

Compute the determinant

Now compute the determinant using the obtained values: \(\text{det}(A) = (1 脳 -1) - (1 脳 1) = -1 - 1 = -2\) The determinant of this given matrix is -2.
03

Calculate the inverse

To get the inverse of a 2x2 matrix, we need to swap the positions of the values a and d, and change the signs of b and c. Hence, the inverse of the matrix A can be expressed as: \[ A^{-1} = \frac{1}{\text{det}(A)} \left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \] Calculate the inverse of the given matrix using the formula and the determinant found in the previous steps.
04

Compute the inverse

Using the values found earlier, compute the inverse matrix A: \[ A^{-1} = \frac{1}{-2} \left[\begin{array}{cc} -1 & -1 \\ -1 & 1 \end{array}\right] = \left[\begin{array}{cc} 1/2 & 1/2 \\ 1/2 & -1/2 \end{array}\right] \] So, the inverse of the given matrix is: \[ \left[\begin{array}{cc} 1/2 & 1/2 \\ 1/2 & -1/2 \end{array}\right] \]
05

Calculate the transpose

To get the transpose of a matrix, we need to switch the rows and columns. In other words, the transpose of the matrix A can be expressed as: \[ A^T = \left[\begin{array}{cc} a & c \\ b & d \end{array}\right] \] Calculate the transpose of the given matrix using the formula.
06

Compute the transpose

Compute the transpose matrix A using the values found earlier: \[ A^T = \left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right] \] So, the transpose of the given matrix is: \[ \left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right] \] As we can see, in this case, the transpose of the matrix is the same as the original matrix.

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Key Concepts

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

Determinant Calculation
The determinant of a matrix is a special number that can provide insights into the matrix itself and the linear transformations it represents. For a 2x2 matrix, the formula to find the determinant is quite simple:
  • For a matrix \( A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \), the determinant is calculated as \( \text{det}(A) = ad - bc \).
  • This represents the area (or volume in higher dimensions) that the matrix transformation scales.
To determine the determinant of the given matrix:
  • Identify the elements: \( a = 1 \), \( b = 1 \), \( c = 1 \), and \( d = -1 \).
  • Substitute into the formula: \( \text{det}(A) = (1 \times -1) - (1 \times 1) \).
  • Calculate the result: \( \text{det}(A) = -1 - 1 = -2 \).
The negative value indicates that the transformation flips the orientation of the area.
Matrix Inversion
Matrix inversion is the process of finding the matrix that acts as a reciprocal or an inverse of a given matrix. When multiplied together, a matrix and its inverse yield the identity matrix, much like how multiplying a number by its reciprocal gives one. For a 2x2 matrix, the inverse formula involves:
  • Swapping the diagonal elements \( a \) and \( d \).
  • Changing the signs of the off-diagonal elements \( b \) and \( c \).
  • Finally, dividing each element by the determinant \( \text{det}(A) \).
To find the inverse of the given matrix:
  • First, confirm that an inverse exists by ensuring the determinant is not zero (which it is not as \( \text{det}(A) = -2 \)).
  • Use the formula: \( A^{-1} = \frac{1}{\text{det}(A)} \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right] \).
  • Substitute the values: \( A^{-1} = \frac{1}{-2} \left[\begin{array}{cc} -1 & -1 \ -1 & 1 \end{array}\right] \).
  • Calculate each element: \( A^{-1} = \left[\begin{array}{cc} 1/2 & 1/2 \ 1/2 & -1/2 \end{array}\right] \).
This matrix serves as the inverse of the original, allowing for reversing the transformations it represents.
Matrix Transposition
Matrix transposition involves flipping a matrix over its diagonal, effectively swapping its rows with columns. This is an operation that results in what is called the transpose of a matrix. For any matrix \( A \), its transpose is denoted as \( A^T \). To transpose a matrix:
  • Each element \( a_{ij} \) becomes \( a_{ji} \), meaning the element that was in row \( i \) and column \( j \) is moved to row \( j \) and column \( i \).
  • This operation doesn't alter the determinant nor the product of the matrix's eigenvalues.
For the example matrix given:
  • The transpose is calculated by switching its rows with its columns.
  • The resulting matrix \( A^T \) appears as \( \left[\begin{array}{cc} 1 & 1 \ 1 & -1 \end{array}\right] \).
Interestingly, in this case, the transpose of the matrix is the same as the original matrix, which indicates a special symmetry property.

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Most popular questions from this chapter

What does it mean when we say that \((A+B)_{i j}=A_{i j}+B_{i j}\) ?

Evaluate the given expression. Take \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]\), and \(C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .\) $$ 3 B^{T} $$

Textbook Writing You are writing a college-level textbook on finite mathematics, and are trying to come up with the best combination of word problems. Over the years, you have accumulated a collection of amusing problems, serious applications, long complicated problems, and "generic" problems. \({ }^{25}\) Before your book is published, it must be scrutinized by several reviewers who, it seems, are never satisfied with the mix you use. You estimate that there are three kinds of reviewers: the "no-nonsense" types who prefer applications and generic problems, the "dead serious" types, who feel that a collegelevel text should be contain little or no humor and lots of long complicated problems, and the "laid-back" types, who believe that learning best takes place in a light-hearted atmosphere bordering on anarchy. You have drawn up the following chart, where the payoffs represent the reactions of reviewers on a scale of \(-10\) (ballistic) to \(+10\) (ecstatic): Reviewers ou \begin{tabular}{|l|c|c|c|} \hline & No-Nonsense & Dead Serious & Laid-Back \\ \hline Amusing & \(-5\) & \(-10\) & 10 \\ \hline Serious & 5 & 3 & 0 \\ \hline Long & \(-5\) & 5 & 3 \\ \hline Generic & 5 & 3 & \(-10\) \\ \hline \end{tabular} a. Your first draft of the book contained no generic problems, and equal numbers of the other categories. If half the reviewers of your book were "dead serious" and the rest were equally divided between the "no-nonsense" and "laid-back" types, what score would you expect? b. In your second draft of the book, you tried to balance the content by including some generic problems and eliminating several amusing ones, and wound up with a mix of which one eighth were amusing, one quarter were serious, three eighths were long, and a quarter were generic. What kind of reviewer would be least impressed by this mix? c. What kind of reviewer would be most impressed by the mix in your second draft?

T o u r i s m ~ i n ~ t h e ~ ' 9 0 s ~ T h e ~ f o l l o w i n g ~ t a b l e ~ g i v e s ~ t h e ~ n u m b e r ~ }\\\ &\text { of people (in thousands) who visited Australia and South }\\\ &\text { Africa in } 1998 .^{13} \end{aligned} You estimate that \(5 \%\) of all visitors to Australia and \(4 \%\) of all visitors to South Africa decide to settle there permanently. Take \(A\) to be the \(3 \times 2\) matrix whose entries are the 1998 tourism figures in the above table and take $$ B=\left[\begin{array}{l} 0.05 \\ 0.04 \end{array}\right] \text { and } C=\left[\begin{array}{ll} 0.05 & 0 \\ 0 & 0.04 \end{array}\right] $$ Compute the products \(A B\) and \(A C\). What do the entries in these matrices represent?

Revenue Recall the Left Coast Bookstore chain from the preceding section. In January, it sold 700 hardcover books, 1,300 softcover books, and 2,000 plastic books in San Francisco; it sold 400 hardcover, 300 softcover, and 500 plastic books in Los Angeles. Now, hardcover books sell for \(\$ 30\) each, softcover books sell for \(\$ 10\) each, and plastic books sell for \(\$ 15\) each. Write a column matrix with the price data and show how matrix multiplication (using the sales and price data matrices) may be used to compute the total revenue at the two stores.
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