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Chapter 9: Problem 5

Find the value of each determinant. $$\left|\begin{array}{rr} 9 & 3 \\ -3 & -1 \end{array}\right|$$

Short Answer

Expert verified
The value of the determinant is 0.

Step by step solution

01

- Understand the Determinant Formula for a 2x2 Matrix

For a 2x2 matrix \(\begin{array}{cc} a & b \ c & d \ \end{array}\), the determinant is calculated as \(ad - bc\).
02

- Identify Elements

In the given matrix \(\begin{array}{cc} 9 & 3 \-3 & -1\ \end{array}\), identify the elements: \(a = 9\), \(b = 3\), \(c = -3\), and \(d = -1\).
03

- Substitute Elements into the Determinant Formula

Substitute the identified elements into the determinant formula: \(det = (9) \cdot (-1) - (3) \cdot (-3)\).
04

- Calculate the Deteminant

Compute the expression: \((-9) + (9) = 0\). The value of the determinant is 0.

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Key Concepts

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

2x2 Matrix
A matrix is an array of numbers organized in rows and columns. A 2x2 matrix specifically has 2 rows and 2 columns. For example, the matrix \(\begin{array}{cc} 9 & 3 \ -3 & -1 \ \ \ \ \end{array}\) is a 2x2 matrix. Each element in the matrix can be identified by its position within the matrix (first row, first column etc.).
Determinant Formula
The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant helps determine if the matrix has an inverse and is vital in solving linear equations. The formula for the determinant of a 2x2 matrix \(\begin{array}{cc} a & b \ c & d \ \ \end{array}\) is given by \(ad - bc\).
Matrix Operations
Matrix operations involve certain procedures that can be performed on matrices, such as addition, subtraction, and multiplication. When working with determinants, the primary operation is substitution and multiplication of matrix elements based on their positions. For example, in our exercise, we have the matrix \(\begin{array}{cc} 9 & 3 \ -3 & -1 \ \ \end{array}\). We identify each element and substitute them into the determinant formula to find the value. We multiply corresponding elements and then subtract the products to get the determinant.
Identifying Matrix Elements
When calculating the determinant of a 2x2 matrix, the first step is to correctly identify each element. For the matrix \(\begin{array}{cc} 9 & 3 \ -3 & -1 \ \ \end{array}\), the elements are: \(
  • \(a = 9\)
  • \(b = 3\)
  • \(c = -3\)
  • \(d = -1\)
\). With these elements identified, you can then substitute them into the determinant formula to complete the calculation.

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Most popular questions from this chapter

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product when possible. $$C A$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} 3 x-2 y+4 z &=1 \\ 4 x+y-5 z &=2 \\ -6 x+4 y-8 z &=-2 \end{aligned}$$

Solve each system for \(x\) and y using Cramer's rule. Assume a and b are nonzero constants. $$\begin{array}{r} x+b y=b \\ a x+y=a \end{array}$$

Use a system of equations to solve each problem. Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((2,3),(-1,0),\) and \((-2,2)\)

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{r} x+y=4 \\ 2 x-y=2 \end{array}$$
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