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Chapter 4: Problem 9

Find the value of each determinant. \(\left|\begin{array}{ll}0 & 3 \\ 1 & 5\end{array}\right|\)

Short Answer

Expert verified
The determinant is -3.

Step by step solution

01

- Understand the Determinant Formula for a 2x2 Matrix

The determinant of a 2x2 matrix \(\begin{array}{ll}a & b \ c & d \end{array} \) is calculated using the formula: \[ \text{Determinant} = ad - bc \]
02

- Identify the Elements of the Matrix

Identify the elements of the given matrix where \( a = 0, b = 3, c = 1, d = 5 \)
03

- Substitute the Elements into the Determinant Formula

Substitute these values into the determinant formula: \[ \text{Determinant} = (0 \times 5) - (3 \times 1) \]
04

- Perform the Calculation

Carry out the multiplication and subtraction: \[ \text{Determinant} = 0 - 3 = -3 \]

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Key Concepts

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

2x2 matrix
Let's start by understanding what a 2x2 matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. A 2x2 matrix specifically has 2 rows and 2 columns. For example, the given matrix for the exercise is: \[ \begin{array}{ll} 0 & 3 \ 1 & 5 \ \ \ \ \ \ \ \ \end{array} \] The numbers inside the matrix are called elements. In this 2x2 matrix, we have four elements: 0, 3, 1, and 5.
matrix operations
Matrix operations include addition, subtraction, multiplication, and finding determinants among others. Here, we focus on finding the determinant. For a 2x2 matrix with elements \(a, b, c, d\), the determinant expresses a specific scalar value derived from these elements. Identifying each element correctly is crucial for any operations. Let鈥檚 identify the elements in the given matrix, which are:
  • a = 0
  • b = 3
  • c = 1
  • d = 5
determinant calculation
Now, let's calculate the determinant of the 2x2 matrix using the formula: \[ \text{Determinant} = ad - bc \] Substituting the identified elements into the formula:
\[ \text{Determinant} = (0 \times 5) - (3 \times 1) \] Now, perform the multiplications: \[ 0 \times 5 = 0 \] \[ 3 \times 1 = 3 \] Subtract the second result from the first: \[ 0 - 3 = -3 \] Therefore, the determinant of the matrix is -3.

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Most popular questions from this chapter

Solve each problem by using a system of three equations in three unknowns. Three coins. Nelson paid 1.75 dollars for his lunch with 13 coins, consisting of nickels, dimes, and quarters. If the number of dimes was twice the number of nickels, then how many of each type of coin did he use?

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