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Chapter 6: Problem 13

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} -6 & 4 \\ -3 & 2 \end{array}\right]$$

Short Answer

Expert verified
Matrix A is not invertible because its determinant is 0.

Step by step solution

01

Determinant Calculation

The formula to find the inverse of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). First, we need to find the determinant of matrix \(A\).The determinant \(det(A)\) is calculated as follows: \(-6 \cdot 2 - (-3) \cdot 4 = -12 + 12 = 0\).
02

Check for Invertibility

A matrix is only invertible if its determinant is not equal to zero. Since the determinant of matrix \(A\) is 0, \(A\) does not have an inverse.

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Key Concepts

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

The Determinant of a Matrix
In the context of matrices, the determinant is a special scalar value that can provide a lot of information about the properties of a matrix. For a 2x2 matrix given by \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\] the determinant, denoted as \( \det(A) \), is calculated using the formula:\[\det(A) = ad - bc.\]
  • The determinant helps us understand whether a matrix is invertible.
  • If the determinant equals zero, the matrix is singular and does not have an inverse.
  • If the determinant is non-zero, the matrix is non-singular, implying it has an inverse.
For example, consider the matrix \[A = \begin{bmatrix} -6 & 4 \ -3 & 2 \end{bmatrix}.\]Calculating its determinant gives \[(-6)(2) - (-3)(4) = -12 + 12 = 0.\]Thus, the determinant is zero, indicating that this particular matrix is not invertible.
Understanding the 2x2 Matrix
A 2x2 matrix is a grid that contains two rows and two columns filled with numbers. It is one of the simplest forms of a matrix, making it a great starting point for learning matrix operations like finding an inverse. The general form of a 2x2 matrix is written as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}.\]
  • Each component of the matrix, \(a, b, c,\) and \(d\), is a real number that affects the matrix鈥檚 properties.
  • Arithmetic operations, like addition, multiplication, and finding the determinant and inverse, become intuitive once you know these elements.
  • These matrices are extensively used in transformations, systems of equations, and linear algebra.
The simplicity of the 2x2 matrix showcases fundamental linear algebra concepts, providing a manageable way to compute determinants and inverses.
Invertibility and Its Significance
The concept of invertibility in matrices ties closely with solving systems of equations and ensuring transformations are reversible. A matrix is invertible if and only if it has an inverse, meaning that there exists another matrix which, when multiplied together, results in an identity matrix.
  • An identity matrix in the 2x2 form is \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix},\]serving as a multiplicative equivalent of the number 1 in basic arithmetic.
  • Invertibility hinges entirely on the determinant: only non-zero determinants allow for an inverse.
  • If a matrix is not invertible, it is considered singular. Operations like matrix division become impossible, making the matrix limited in its applications.
Understanding whether a matrix is invertible is thus crucial in fields such as engineering, physics, and computer graphics, where matrix operations model real-world data.

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Most popular questions from this chapter

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$\left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the \(3 \times 3\) matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A circle and a parabola; four points.

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{array}{r}3 x-y=4 \\\x+y=0\end{array}$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{c} x+y \leq 4 \\ x-y \leq 5 \\ 4 x+y \leq-4 \end{array}$$

The relationship between a professional basketball player's height \(h\) in inches and weight \(w\) in pounds was modeled by using two samples of players. The resulting equations were $$\begin{aligned}&w=7.46 h-374\\\&w=7.93 h-405\end{aligned}$$ and Assume that \(65 \leq h \leq 85\) (a) Use each equation to predict the weight to the nearest pound of a professional basketball player who is 6 feet 11 inches. (b) Determine graphically the height at which the two models give the same weight. (c) For each model, what change in weight is associated with a 1 -inch increase in height?
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