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Chapter 6: Problem 4

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right] ; B=\left[\begin{array}{rr} 2 & 1 \\ -3 & 2 \end{array}\right]$$

Short Answer

Expert verified
A and B are not inverses because their products AB and BA do not equal the identity matrix.

Step by step solution

01

Calculate Matrix Product AB

To determine if matrices \( A \) and \( B \) are inverses, we first compute the product \( AB \). The matrices are given as \( A = \begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 1 \ -3 & 2 \end{bmatrix} \).To multiply the matrices \( AB \):1. Compute the element in the first row and first column: \( (2)(2) + (1)(-3) = 4 - 3 = 1 \).2. Compute the element in the first row and second column: \( (2)(1) + (1)(2) = 2 + 2 = 4 \).3. Compute the element in the second row and first column: \( (3)(2) + (2)(-3) = 6 - 6 = 0 \).4. Compute the element in the second row and second column: \( (3)(1) + (2)(2) = 3 + 4 = 7 \).Thus, \( AB = \begin{bmatrix} 1 & 4 \ 0 & 7 \end{bmatrix} \).
02

Calculate Matrix Product BA

Next, we compute the product \( BA \):Use the given matrices \( B = \begin{bmatrix} 2 & 1 \ -3 & 2 \end{bmatrix} \) and \( A = \begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix} \).1. Compute the element in the first row and first column: \( (2)(2) + (1)(3) = 4 + 3 = 7 \).2. Compute the element in the first row and second column: \( (2)(1) + (1)(2) = 2 + 2 = 4 \).3. Compute the element in the second row and first column: \( (-3)(2) + (2)(3) = -6 + 6 = 0 \).4. Compute the element in the second row and second column: \( (-3)(1) + (2)(2) = -3 + 4 = 1 \).Thus, \( BA = \begin{bmatrix} 7 & 4 \ 0 & 1 \end{bmatrix} \).
03

Determine if A and B are Inverses

Finally, recall that two matrices are inverses of each other if their product in both directions is the identity matrix, which is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).- The product \( AB = \begin{bmatrix} 1 & 4 \ 0 & 7 \end{bmatrix} \) is not the identity matrix.- Similarly, the product \( BA = \begin{bmatrix} 7 & 4 \ 0 & 1 \end{bmatrix} \) is not the identity matrix.Since neither \( AB \) nor \( BA \) is the identity matrix, matrices \( A \) and \( B \) are not inverses.

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Key Concepts

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

Matrix Multiplication
Matrix multiplication is an essential operation in linear algebra. It involves the process of combining two matrices to produce a third matrix. To multiply two matrices, we follow specific rules:
  • Each element in the resulting matrix is computed by taking the dot product of the rows of the first matrix and the columns of the second matrix.
  • An element in position (i, j) of the resulting matrix is obtained by multiplying the corresponding rows and columns and then adding their products.
  • The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.
To illustrate, consider matrices A and B. If matrix A is of size 2x2 and matrix B is also 2x2, they can be multiplied to produce a matrix of the same size (2x2). These matrices are multiplied step-by-step, focusing on row and column combinations, which emphasizes understanding over rote computation.
This structured approach helps ensure calculations are precise and helps students better grasp the concept of matrix multiplication.
Identity Matrix
The identity matrix is a fundamental concept in the study of matrices, acting as the '1' of matrix arithmetic. It plays a crucial role because it acts as the neutral element in matrix multiplication.
  • For a square matrix, the identity matrix is a matrix with ones on the diagonal and zeros elsewhere. For example, a 2x2 identity matrix is represented as \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).
  • When you multiply any matrix by the identity matrix (on either side), the original matrix remains unchanged.
  • The identity property holds true only when the identity matrix and the matrix being multiplied are of compatible sizes.
The concept of the identity matrix is essential, especially when determining if two matrices are inverses. If the product of two matrices results in the identity matrix, it indicates that they are inverses of each other. This characteristic is key in theoretical and practical applications in matrix algebra.
Determinants
Determinants are numerical values derived from square matrices and are pivotal in defining important matrix properties such as invertibility. They provide insights into whether a matrix can be inverted or not.
  • For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as: \( ad - bc \).
  • A matrix can only have an inverse if its determinant is non-zero. A determinant of zero indicates the matrix is singular, which means it does not have an inverse.
  • Determinants can also help assess the scalability of transformations represented by matrices in geometry and other fields.
Understanding determinants bridges the gap between theoretical matrix concepts and practical computations, allowing students to predict matrix behavior in equations and systems. They serve as a foundational step in problems related to matrix inverses and are useful for simplifying complex matrix algebra.

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Most popular questions from this chapter

In certain parts of the Rocky Mountains, deer are the main food source for mountain lions. When the deer population \(d\) is large, the mountain lions ( \(m\) ) thrive. However, a large mountain lion population drives down the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$\left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{cc} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 year? 2 years? (c) Consider part (b), but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of \(1 \$ 6\).

Solve each application. Financing Expansion To get funds necessary for a planned expansion, a small company took out three loans totaling \(\$ 12,500 .\) The company was able to borrow some of the money at \(2 \% .\) It borrowed \(\$ 1000\) more than \(\frac{1}{2}\) the amount of the \(2 \%\) loan at \(3 \%\) and the rest at \(2.5 \% .\) The total annual interest was \(\$ 305 .\) How much did the company borrow at each rate?

Farmer Jones raises only pigs and geese. She wants to raise no more than 16 animals, with no more than 12 geese. She spends \(\$ 50\) to raise a pig and \(\$ 20\) to raise a goose. She has \(\$ 500\) available for this purpose. Find the maximum profit she can make if she makes a profit of \(\$ 80\) per goose and \(\$ 40\) per pig. Indicate how many pigs and geese she should raise to achieve this maximum.

Solve each system of four equations in four variables. Express the solutions in the form \((x, y, z, w)\) $$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\\ x-y-3 z-2 w &=2 \end{aligned}$$

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. (Refer to Exercise 92 .) Suppose that supply is related to price by \(p=\frac{1}{10} q\) and that demand is related to price by \(p=15-\frac{2}{3} q,\) where \(p\) is price in dollars and \(q\) is the quantity supplied in units. (a) Determine the price at which 15 units would be supplied. Determine the price at which 15 units would be demanded. (b) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?
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