Problem 56 - Algebra 2

Question

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{6} & {4} \\ {-3} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 24.

Step 1: Recall the formula for a 2x2 determinant

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula \( ad - bc \). This formula takes the product of the diagonal elements and subtracts the product of the off-diagonal elements.

Step 2: Identify the elements of the matrix

For the given matrix \( \begin{bmatrix} 6 & 4 \ -3 & 2 \end{bmatrix} \), identify the elements: \( a = 6 \), \( b = 4 \), \( c = -3 \), and \( d = 2 \).

Step 3: Apply the determinant formula

Substitute the identified values into the determinant formula: \( ad - bc \). So, we compute \( (6)(2) - (4)(-3) \).

Step 4: Perform the calculations

Calculate the products: \( 6 \times 2 = 12 \) and \( 4 \times -3 = -12 \). Then apply these in the formula, which becomes \( 12 - (-12) \).

Step 5: Simplify the expression

Simplify \( 12 - (-12) \) to get \( 24 \). This is the determinant of the matrix.

Key Concepts

Understanding 2x2 Matrices

When diving into matrices, an important type we often come across is the 2x2 matrix. It's a simple structure used in many mathematical computations. A 2x2 matrix has two rows and two columns, written as follows:

• The top row comprises two entries. We denote these as \( a \) and \( b \).
• The bottom row consists of \( c \) and \( d \).

Thus, the matrix appears as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Each entry holds crucial information about the matrix itself. The arrangement into rows and columns helps us to easily apply processes, like finding the determinant, which is a key part of matrix analysis. That's why becoming comfortable with identifying different parts of a 2x2 matrix is essential for moving forward in linear algebra.

Performing Matrix Calculations

Matrix calculations form the backbone of many operations in linear algebra, especially when determining properties like the determinant. Calculating the determinant of a 2x2 matrix involves a specific arithmetic procedure that can be memorized and easily performed. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula for the determinant is:

• Calculate the product of the diagonal elements: \( a \times d \).
• Next, find the product of the off-diagonal elements: \( b \times c \).
• Subtract the off-diagonal product from the diagonal product to get the determinant: \( ad - bc \).

This simple calculation helps us understand how the elements in the matrix interact with each other, providing insights into the matrix's characteristics. It's a small computation, but crucial for solving problems like those in physics, engineering, and more advanced mathematics.

Important Concepts in Linear Algebra

Linear algebra is a branch of mathematics dedicated to the study of vectors, matrices, and linear transformations. It extends the basic operations performed in 2x2 matrices to wider contexts. Key concepts include:

• Vectors: These are entities that have both magnitude and direction, used to describe points in space.
• Matrices: Collections of numbers arranged into rows and columns, used to transform vectors through operations.
• Linear Transformations: Operations that map vectors to other vectors in a way that preserves vector addition and scalar multiplication.

The determinant, as explored with matrices, is a tool for understanding these transformations, indicating things like area scaling and invertibility. Mastery of linear algebra lays down the foundation for exploring systems of equations, optimizations, and much more.