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Chapter 6: Problem 16

Find each determinant. $$\operatorname{det}\left[\begin{array}{ccc}2 & 1 & -1 \\\4 & 7 & -2 \\\2 & 4 & 0\end{array}\right]$$

Short Answer

Expert verified
The determinant is 10.

Step by step solution

01

Write the matrix and formula

We need to find the determinant of a 3x3 matrix. The matrix is given by \(A = \begin{bmatrix} 2 & 1 & -1 \ 4 & 7 & -2 \ 2 & 4 & 0 \end{bmatrix}\). The formula for the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is \( \operatorname{det}(A) = a(ei鈭抐h) - b(di鈭抐g) + c(dh鈭抏g) \).
02

Identify elements of the matrix

Identify each of the elements in the determinant formula: \( a = 2, b = 1, c = -1, d = 4, e = 7, f = -2, g = 2, h = 4, i = 0 \).
03

Substitute elements into the formula

Substitute the elements into the determinant formula:\[\operatorname{det}(A) = 2(7 \cdot 0 - (-2) \cdot 4) - 1(4 \cdot 0 - (-2) \cdot 2) + (-1)(4 \cdot 4 - 7 \cdot 2)\].
04

Simplify calculations inside parentheses

Calculate each of the products inside the determinants:- \(7 \cdot 0 = 0\) and \((-2) \cdot 4 = -8\), so \(7 \cdot 0 - (-2) \cdot 4 = 8\).- \(4 \cdot 0 = 0\) and \((-2) \cdot 2 = -4\), so \(4 \cdot 0 - (-2) \cdot 2 = 4\).- \(4 \cdot 4 = 16\) and \(7 \cdot 2 = 14\), so \(4 \cdot 4 - 7 \cdot 2 = 2\).
05

Compute final determinant value

Now substitute back into the simplified expression:\[\operatorname{det}(A) = 2 \cdot 8 - 1 \cdot 4 + (-1) \cdot 2 = 16 - 4 - 2 = 10\].
06

Conclusion

The determinant of the matrix \(A\) is \(10\). Hence, \(\operatorname{det}\left(\begin{bmatrix} 2 & 1 & -1 \ 4 & 7 & -2 \ 2 & 4 & 0 \end{bmatrix}\right) = 10\).

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Key Concepts

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

Understanding a 3x3 Matrix
A 3x3 matrix is a simple type of square matrix composed of three rows and three columns. Matrices are arrays of numbers used to organize data or equations. In a 3x3 matrix, there are nine elements represented typically as letters based on their position.- Consider a generic matrix: \ \[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\] Each letter from 'a' to 'i' corresponds to different positions within the matrix.- The goal when working with matrices often involves finding the determinant.- Determining characteristics like invertibility or solvability of systems often depend on this value.In our exercise, the matrix given is:\[\begin{bmatrix}2 & 1 & -1 \4 & 7 & -2 \2 & 4 & 0\end{bmatrix}\] Each specific number represents unique data values that can be worked with through matrix operations.
Determinant Calculation Made Easy
The determinant of a matrix is a special number that can provide a lot of information about the matrix. For a 3x3 matrix, a determinant is calculated using a specific formula which involves the values present in the matrix's elements.- The basic steps to calculate the determinant of a 3x3 matrix, given: \[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i\end{bmatrix}\] - Start by identifying each element (a to i). - Use the formula: - \(\operatorname{det}(A) = a(ei鈭抐h) - b(di鈭抐g) + c(dh鈭抏g)\) - Substitute each matrix element into the formula. - Simplify the calculations to find the determinant.This determinant can inform if a matrix is invertible (non-zero determinant) or singular (if determinant is zero). Here, simpler operations inside the parentheses such as multiplication and subtraction dictate steps of simplification.
A Brief Overview of Matrix Algebra
Matrix algebra is a branch of mathematics that deals with matrices and their operations. It is widely used in various fields including computer science, economics, and engineering. - **Key Operations:** - Addition and subtraction: Requires matrices of the same dimensions. - Scalar multiplication: Involves multiplying each matrix element by a scalar (number). - Matrix multiplication: More complex, not commutative, order matters. - **Role of Determinants:** - Helps determine if matrices have inverses. - Used in solving systems of linear equations through methods such as Cramer's Rule. Learning matrix algebra empowers one to handle problems involving data structures. It becomes easier to transform systems of equations into simpler forms for solving real-world problems. Calculating determinants is a foundational skill, aiding in uncovering these deeper aspects of matrix math.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \leq \log x\\\ &y \geq|x-2| \end{aligned}$$

A wholesaler of party goods wishes to display her products at a convention of social secretaries in such a way that she gets the maximum number of inquiries about her whistles and hats. Her booth at the convention has 12 square meters of floor space to be used for display purposes. A display unit for hats requires 2 square meters, and one for whistles requires 4 square meters. Experience tells the wholesaler that she should never have more than a total of 5 units of whistles and hats on display at one time. If she receives three inquiries for each unit of hats and two inquiries for each unit of whistles on display, how many of each should she display in order to get the maximum number of inquiries? What is that maximum number?

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rr} \frac{3}{20} & \frac{1}{4} \\ -\frac{1}{20} & \frac{1}{4} \end{array}\right]$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} 2 x+3 y & \leq 12 \\ 2 x+3 y & > -6 \\ 3 x+y & < 4 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}$$

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((A B) C=A(B C)\) (associative property)
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