Problem 55 - Algebra 2

Question

Evaluate each determinant. $$ \left|\begin{array}{rr}{3} & {0} \\ {2} & {-2}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -6.

Step 1: Identify the Determinant Formula

The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula \( ad - bc \). In this matrix, \( a = 3 \), \( b = 0 \), \( c = 2 \), and \( d = -2 \).

Step 2: Substitute Values into the Formula

Use the determinant formula \( ad - bc \) with the identified values: \( 3 \times (-2) - 0 \times 2 \).

Step 3: Calculate the Products

Calculate each part individually: \( 3 \times (-2) = -6 \) and \( 0 \times 2 = 0 \).

Step 4: Solve the Determinant Formula

Subtract the second product from the first as per the determinant formula: \( -6 - 0 = -6 \).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple yet fundamental concept in linear algebra. It is essentially a grid with two rows and two columns. Imagine it like a small table where each cell contains a number. For example, the matrix \[ \begin{bmatrix} 3 & 0 \ 2 & -2 \end{bmatrix} \] is a 2x2 matrix, where:

• "3" is in the first row, first column (top left position)
• "0" is in the first row, second column (top right position)
• "2" is in the second row, first column (bottom left position)
• "-2" is in the second row, second column (bottom right position)

These matrices are commonly used to represent systems of linear equations, transformations, and other mathematical operations. The simplicity of the 2x2 matrix makes it ideal for learning how different computations and operations are performed in linear algebra.

Determinant Formula for a 2x2 Matrix

The determinant of a matrix is a special number that can be calculated from its elements. Specifically, for a 2x2 matrix, the determinant helps in understanding certain properties of the matrix, like whether it is invertible. The formula to compute the determinant \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is straightforward:\[ ad - bc \]Here,

• "a" and "d" are the elements on the main diagonal of the matrix
• "b" and "c" are the off-diagonal elements

To find the determinant, multiply the diagonal elements 'a' and 'd', then subtract the product of the off-diagonal elements 'b' and 'c'. For the given matrix:\[ \begin{pmatrix} 3 & 0 \2 & -2 \end{pmatrix} \]we substitute the values: \( ad - bc = 3(-2) - 0(2) \) and compute it to get \( -6 \). The determinant tells us multiple things about the matrix like how it can scale space and if it has an inverse.

Basic Concepts of Matrix Operations

Matrix operations are fundamental tools used to manipulate matrices in various ways. Here are some common operations you might encounter when working with matrices:

• Addition: To add two matrices of the same size, simply add their corresponding elements.
• Subtraction: Similarly, to subtract one matrix from another, subtract their corresponding elements.
• Multiplication: You can multiply matrices too, but matrix multiplication is not as simple as adding. It involves taking the dot product of rows and columns.
• Inverse: Finding the inverse of a matrix (if it exists) is a bit more complex. For a 2x2 matrix, the inverse can be found if the determinant is not zero.

In the case of 2x2 matrices, these operations often serve as the building blocks for more advanced computations in mathematics, physics, and engineering. Understanding matrix operations allows you to work on complex problems involving systems of equations, transformations, and much more.