Problem 15 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{ll} -2 & -3 \\ -4 & -5 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -2.

Step 1: Identify Determinant Formula for 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). This formula applies to any matrix of this size.

Step 2: Assign Matrix Values

Assign the given matrix values to variables: \( a = -2 \), \( b = -3 \), \( c = -4 \), and \( d = -5 \).

Step 3: Apply Determinant Formula

Substitute the values into the determinant formula: \( (-2)(-5) - (-3)(-4) \).

Step 4: Calculate Each Term

Calculate \( (-2)(-5) = 10 \) and \( (-3)(-4) = 12 \).

Step 5: Subtract Second Product from First

Subtract the second product from the first: \( 10 - 12 = -2 \).

Step 6: State the Final Determinant

The determinant of the matrix is \(-2\).

Key Concepts

Understanding a 2x2 Matrix

Matrices are a way to organize numbers in a grid of rows and columns. A 2x2 matrix is a small matrix that consists of two rows and two columns. For example, in this exercise, the matrix is written as : \[ \begin{bmatrix} -2 & -3 \ -4 & -5 \end{bmatrix} \]
This matrix has four elements:

• -2 and -3 in the first row
• -4 and -5 in the second row

Identifying these elements is the first step in many matrix operations. They are essential for operations like addition, multiplication, or finding the determinant. In our case, the focus is on understanding and calculating its determinant, which is a single value representing specific properties of the matrix. A 2x2 matrix is quite straightforward and serves as a good starting point for learning about matrices and their determinants.

Introduction to Matrix Algebra

Matrix algebra involves mathematical operations that can be performed on matrices. These operations include addition, subtraction, multiplication, and finding the determinant.
Each operation has specific rules. For 2x2 matrices, operations are particularly simpler, which makes them great for beginners to practice matrix algebra.
When it comes to finding determinants in matrix algebra:

• It involves a specific formula that translates a matrix into a single number.
• This number can reveal information about the matrix, like whether it has an inverse.

Understanding matrix algebra and its operations is fundamental when dealing with systems of equations. Such systems can often be represented and solved using matrices. It's a key concept in fields like physics, computer graphics, and any area involving linear transformations.

Simple Algebraic Calculation of Determinants

Calculating the determinant of a 2x2 matrix is an algebraic process that uses a simple formula: \[\text{Determinant} = ad - bc\]This formula takes the elements of the matrix and combines them through multiplication and subtraction.
For the example matrix:

• First, multiply the top-left and bottom-right elements (\(-2\times -5\) = 10).
• Then, multiply the top-right and bottom-left elements (\(-3\times -4\) = 12).
• Finally, subtract the second product from the first (\(10 - 12\) = -2).

This simple calculation provides us with the determinant: -2. This number can tell us about the matrix's properties. If a determinant is zero, it indicates the matrix is singular and does not have an inverse. Thus, such calculations play an essential role in matrix algebra and broader mathematical applications.