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Chapter 10: Problem 19

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 0 & -2 & 4 \\ 0 & 1 & -3 \end{array}\right]$$

Short Answer

Expert verified
The determinant is 4; the matrix is invertible.

Step by step solution

01

Identify the Matrix

The given matrix is a 3x3 matrix:\[A = \begin{pmatrix} 2 & 1 & 0 \ 0 & -2 & 4 \ 0 & 1 & -3 \end{pmatrix}\]
02

Calculate the Determinant

To find the determinant of a 3x3 matrix, we use the formula:\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]where the matrix elements are as follows:\[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\]Substituting the given matrix elements:\[ a = 2, \, b = 1, \, c = 0, \, d = 0, \, e = -2, \, f = 4, \, g = 0, \, h = 1, \, i = -3 \]The determinant is:\[\text{det}(A) = 2((-2)(-3) - (4)(1)) - 1((0)(-3) - (4)(0)) + 0((0)(1) - (-2)(0))\]This simplifies to:\[= 2(6 - 4) - 1(0 - 0) + 0(0 - 0)\]\[= 2 \times 2 - 0 + 0 = 4\]
03

Determine Invertibility of the Matrix

A matrix is invertible if its determinant is not zero. The calculated determinant is 4, which is not zero. Therefore, the matrix is invertible.

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Key Concepts

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

Invertibility
A matrix is said to be invertible if there exists another matrix that, when multiplied with the original matrix, results in the identity matrix. This concept is fundamental in matrix algebra since it mirrors the idea of division for real numbers.
For a matrix to have an inverse, a crucial requirement is that its determinant must not be zero.
In simpler terms, if the determinant of a matrix equals zero, the matrix cannot be inverted. But, if the determinant is not zero, you can potentially find the inverse matrix.
This property is useful in solving systems of linear equations or in transforming a system into another form in linear algebra.
  • Only square matrices can be invertible, meaning the number of rows equals the number of columns.
  • The inverse matrix is a powerful tool in various fields such as computer graphics, cryptography, and statistics.
3x3 Matrix
A 3x3 matrix consists of three rows and three columns, making it a square matrix. This type of matrix is common in linear algebra problems due to its balance between complexity and manageability.
Understanding a 3x3 matrix involves identifying all nine elements within the structure.
These elements are crucial when performing operations such as calculating a determinant or finding an inverse.
  • Each element in a 3x3 matrix has a specific position, denoted by its row and column.
  • Identifying the layout of the matrix is the first step in performing mathematical operations like determinant calculation.

A 3x3 matrix can represent various systems, from vector transformations to equations in three-dimensional space, making it a versatile mathematical tool.
Determinant Calculation
The determinant is a special value that can be calculated from a square matrix, providing insight into properties like invertibility and volume transformation.
For a 3x3 matrix, the determinant can be computed using a specific formula, involving each element of the matrix.
The formula for a 3x3 matrix is: \[ ext{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]where each variable represents specific positions in the matrix.
This computation involves multiplying and subtracting various products of the matrix elements.
  • A positive determinant indicates a certain type of linear transformation, whereas a negative value indicates orientation change.
  • Understanding the calculation process helps in applications like determining solvability of matrix equations or assessing coordinate systems.
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations and matrices. It is foundational for understanding spaces and transformations, fundamental in physics, computer science, and engineering.
Within linear algebra, matrices serve as a way to represent and solve linear systems.
Concepts like determinants and invertibility are central to many linear algebra applications.
  • Linear systems arising from real-world problems can often be modeled with matrices and solved using linear algebra techniques.
  • Knowledge of linear algebra is essential for fields that utilize data transformations, machine learning, and graphics.
  • Techniques in linear algebra allow for transformations such as rotations, reflections, and scalings in higher dimensions.

Mastering the basics of linear algebra opens doors to advanced studies and practical applications in diverse scientific and engineering disciplines.

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Most popular questions from this chapter

Find the inverse of the matrix if it exists. $$\left[\begin{array}{llll} 1 & 2 & 0 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 2 & 0 & 2 \end{array}\right]$$

To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$\left\\{\begin{aligned} x+2 y &>4 \\ -x+y &<1 \\ x+3 y &<9 \\ x &<3 \end{aligned}\right.$$

A roadside fruit stand sells apples at \(75 \notin\) a pound, peaches at \(90 \notin\) a pound, and pears at \(60 \notin\) a pound. Muriel buys 18 pounds of fruit at a total cost of 13.80 dollar Her peaches and pears together cost 1.80 dollar more than her apples. (a) Set up a linear system for the number of pounds of apples, peaches, and pears that she bought. (b) Solve the system using Cramer's Rule.

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l} y \leq 9-x^{2} \\ x \geq 0, \quad y \geq 0 \end{array}\right.$$

An encyclopedia saleswoman works for a company that offers three different grades of bindings for its encyclopedias: standard, deluxe, and leather. For each set that she sells, she earns a commission based on the set's binding grade. One week she sells one standard, one deluxe, and two leather sets and makes \(\$ 675\) in commission. The next week she sells two standard, one deluxe, and one leather set for a \(\$ 600\) commission. The third week she sells one standard, two deluxe, and one leather set, earning \(\$ 625\) in commission. (a) Let \(x, y,\) and \(z\) represent the commission she earns on standard, deluxe, and leather sets, respectively. Translate the given information into a system of equations in \(x, y\) and \(z\) (b) Express the system of equations you found in part (a) as a matrix equation of the form \(A X=B\). (c) Find the inverse of the coefficient matrix \(A\) and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on a set of encyclopedias in each grade of binding?
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