Problem 2 - Algebra and Trigonometry

Question

\(1-8\) 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 of the matrix is 2.

Step 1: Understand the Matrix

We are given a 2x2 matrix: \(\begin{bmatrix} 0 & -1 \ 2 & 0 \end{bmatrix}\). To find the determinant, we need to apply the formula specifically for 2x2 matrices.

Step 2: Recall the 2x2 Determinant Formula

The formula for finding the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(ad - bc\). We will use this to calculate the determinant for our matrix.

Step 3: Identify the Elements in the Matrix

In our matrix \(\begin{bmatrix} 0 & -1 \ 2 & 0 \end{bmatrix}\), the elements of the matrix are: \(a = 0\), \(b = -1\), \(c = 2\), \(d = 0\).

Step 4: Substitute Values into the Formula

Substitute those values into the determinant formula: \(ad - bc = (0)(0) - (-1)(2)\).

Step 5: Perform the Calculation

Calculate the expression: \(0 \times 0 = 0\) and \((-1) \times 2 = -2\). So, \(ad - bc = 0 - (-2)\).

Step 6: Final Calculation

Simplify the expression: \(0 - (-2) = 2\). Thus, the determinant of the matrix is 2.

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple yet essential concept in linear algebra. It is a rectangular array consisting of two rows and two columns. Each element within this structure is denoted by a specific position, which plays a crucial role in matrix operations like determining the matrix's determinant. The general representation of a 2x2 matrix is \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] where each element (\(a, b, c, \) and \(d\)) occupies a fixed position in the array.

• The element \(a\) is located in the first row, first column.
• The element \(b\) sits in the first row, second column.
• The element \(c\) is in the second row, first column.
• The element \(d\) holds the second row, second column position.

This compact structure makes calculations, like finding determinants, easier and more straightforward. Learning how to manipulate such matrices is fundamental to solving more complex mathematical problems.

Matrix Elements and Their Roles

The components of a 2x2 matrix, known as matrix elements, are vital for any matrix operation you perform. In our given 2x2 matrix,\[ \begin{bmatrix} 0 & -1 \ 2 & 0 \end{bmatrix} \],each number represents a matrix element:

• \(a = 0\)
• \(b = -1\)
• \(c = 2\)
• \(d = 0\)

These elements are instrumental when calculating the determinant of the matrix. The determinant itself is a unique value that can indicate if a matrix has an inverse, alongside providing other characteristic properties.
The layout of matrix elements defines how they interact in arithmetic operations. Specifically, for finding a determinant, each element's position directly influences the calculation and eventual outcome of matrix processes.

Performing Matrix Operations

Matrix operations involve various manipulative tactics to extract meaningful information or transform the matrix. In the case of a 2x2 matrix, one of the most fundamental operations is finding the determinant.
To do this, utilize the determinant formula for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), which is \(ad - bc\). Here's how you proceed with matrix operations:

• Substitute: Insert the identified elements from the matrix into the formula. For the given matrix, substitute \(a = 0, b = -1, c = 2, d = 0\).
• Calculate: Perform the arithmetic step, compute \(ad - bc\). In our example, this results in \((0)(0) - (-1)(2) = 0 + 2 = 2\).

The result of these operations, a determinant of \(2\), tells you about the matrix’s properties, like its invertibility and stability. Mastery of these operations is key to advancing in topics involving matrices.