Problem 22 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{rr} 3 & -2 \\ 12 & -8 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Identify the matrix elements

The given 2x2 matrix is \( \begin{bmatrix} 3 & -2 \ 12 & -8 \end{bmatrix} \). Let \( a = 3 \), \( b = -2 \), \( c = 12 \), and \( d = -8 \), where these are the elements of the matrix arranged as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).

Step 2: Recall the formula for determinant of a 2x2 matrix

The determinant \( D \) of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula: \[ D = ad - bc \]

Step 3: Substitute the matrix elements into the formula

Substitute the values \( a = 3 \), \( b = -2 \), \( c = 12 \), and \( d = -8 \) into the determinant formula: \[ D = (3)(-8) - (-2)(12) \]

Step 4: Calculate each term in the expression

Calculate \( (3)(-8) = -24 \) and \( (-2)(12) = -24 \).

Step 5: Compute the determinant

Now subtract these results according to the formula: \[ D = -24 - (-24) = -24 + 24 = 0 \]

Step 6: Conclusion

Thus, the determinant of the matrix is 0.

Key Concepts

Matrix Algebra

Matrix algebra provides the framework for manipulating matrices, which are arrays of numbers or variables arranged in rows and columns. Matrices are used to solve linear algebra problems, represent systems of linear equations, and perform transformations in various fields such as physics and engineering. Understanding matrix algebra involves:

• Recognizing the structure of matrices, such as their sizes (e.g., 2x2 for two rows and two columns).
• Identifying the components of a matrix, known as elements, which are usually denoted by variables like \(a, b, c,\) and \(d\) in a 2x2 matrix.
• Performing operations such as addition, subtraction, multiplication, and finding inverses.

In this context, finding the determinant is a vital operation that helps in understanding other properties of a matrix. This operation serves as a key test for matrix invertibility, an important concept in solving systems of equations.

Calculation of Determinants

When we talk about the determinant of a 2x2 matrix, it refers to a scalar value that is derived from the elements of the matrix. The determinant provides valuable information about the matrix. For example, in a system of linear equations, a non-zero determinant implies a unique solution. Here's how you compute it:For a matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant \(D\) is calculated as:\[D = ad - bc\]This formula involves multiplying the top-left and bottom-right elements \(a\) and \(d\) and subtracting the product of the top-right and bottom-left elements \(b\) and \(c\).Example: Consider the matrix given in the problem: \[\begin{bmatrix} 3 & -2 \ 12 & -8 \end{bmatrix}\]Substituting into the formula gives:\[D = (3)(-8) - (-2)(12) = -24 + 24 = 0\]The zero determinant indicates that the matrix is singular, meaning it does not have an inverse.

Intermediate Algebra

Intermediate algebra serves as a bridge between basic algebra and more advanced fields like calculus and linear algebra. It includes understanding how to work with polynomials, rational expressions, and radicals, along with matrices and determinants. Mastering these operations is crucial for solving more complex problems. In this exercise, we focused on calculating the determinant of a given 2x2 matrix. This task involves:

• Identifying the correct matrix elements according to their positions.
• Applying the determinant formula accurately.
• Performing arithmetic operations correctly to achieve the final result.

Intermediate algebra reinforces these skills by encouraging practice with a variety of number types and operations. This aids students in building confidence in manipulating algebraic expressions and solving equations. By understanding the calculation of determinants, students can also gain insights into other algebraic concepts such as matrix invertibility and solutions to linear systems.