Problem 6 - Precalculus Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 0 & -1 \\ 2 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 2.

Step 1: Identify the Matrix

The matrix given is a 2x2 matrix: \[ A = \begin{bmatrix} 0 & -1 \ 2 & 0 \end{bmatrix} \] To find its determinant, we need to apply the determinant formula for a 2x2 matrix.

Step 2: Apply the Determinant Formula

The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is: \[ ext{det}(A) = ad - bc \] For the given matrix, \( a = 0 \), \( b = -1 \), \( c = 2 \), and \( d = 0 \).

Step 3: Calculate the Determinant

Substitute the values into the determinant formula:\[ ext{det}(A) = (0)(0) - (-1)(2) = 0 + 2 = 2 \]

Step 4: Conclusion

The determinant of the matrix \( \begin{bmatrix} 0 & -1 \ 2 & 0 \end{bmatrix} \) is 2.

Key Concepts

Matrices

Matrices are like a magic tool in mathematics, especially useful when dealing with complex, multi-dimensional data. A matrix is essentially a rectangular array of numbers arranged in rows and columns. They can have different sizes, indicated by the number of rows and columns they possess. For example, a matrix with 3 rows and 2 columns is a 3x2 matrix. Matrices are used in various fields such as computer graphics, statistics, and physics, amongst others.

• Rows and Columns: The first number in the size description indicates rows, the second indicates columns.
• Elements: Each entry in a matrix is referred to as an element.

Consider matrices as a way of organizing information in a structured form that is easy to manipulate in mathematical calculations.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, consisting of 2 rows and 2 columns. You can think of it as a small square grid containing four numbers, each occupying a unique position within this grid. Here is an example:\[\begin{bmatrix}a & b \c & d \\end{bmatrix}\]In a 2x2 matrix like this, each number is vital because they play specific roles in calculations involving the matrix, such as finding its determinant. The determinant is an important scalar value that can be computed from the elements of a square matrix, giving insight into the matrix's properties. Determinants are especially easy to calculate with 2x2 matrices as they require only simple arithmetic operations.

Matrix Operations

Matrix operations are fundamental techniques used to manipulate matrices in various ways. With matrices, you can perform a multitude of operations, such as addition, subtraction, multiplication, and finding the determinant.

• Addition and Subtraction: To add or subtract matrices, they must be of the same dimension. Simply add or subtract the corresponding elements.
• Multiplication: Multiplying matrices involves taking the sum of the products of rows and columns. This operation has a rule: the number of columns in the first matrix must equal the number of rows in the second matrix.
• Determinant: For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using \( ad - bc \). This is an important operation that provides insights into the matrix's characteristics.

Understanding these operations are crucial for anyone working in fields that involve mathematics, as they form the basis of many algorithms and solve linear equations efficiently.