Problem 8 - College Algebra

Question

For the following exercises, find the determinant. \(\left|\begin{array}{ll}-8 & 4 \\ -1 & 5\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is -36.

Step 1: Identify the 2x2 Matrix

The problem provides the 2x2 matrix as \( \left[ \begin{array}{cc} -8 & 4 \ -1 & 5 \end{array} \right] \). In general, a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) requires specific elements to multiply and subtract to find the determinant.

Step 2: Apply the Determinant Formula for a 2x2 Matrix

The formula for finding the determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is given by: \( \text{det} = ad - bc \). For our matrix, identify \(a = -8\), \(b = 4\), \(c = -1\), and \(d = 5\).

Step 3: Calculate the Products

Using the values from our matrix, calculate the products: \(ad = (-8)(5) = -40\) and \(bc = (4)(-1) = -4\).

Step 4: Compute the Determinant

Subtract the product \(bc\) from the product \(ad\) using the determinant formula: \(\text{det} = -40 - (-4)\). Simplify the expression to get \(\text{det} = -40 + 4 = -36\).

Step 5: Final Result

The determinant of the given matrix is \(-36\).

Key Concepts

2x2 Matrix

A 2x2 matrix is one of the most basic forms of matrices used in linear algebra. It consists of two rows and two columns. Each element of the matrix is denoted by a unique position within this framework. For example, in a matrix:\[\begin{bmatrix}a & b \c & d \\end{bmatrix},\] 'a' and 'b' represent the elements in the first row, while 'c' and 'd' are the elements in the second row.

The order in which we arrange a 2x2 matrix or any matrix is very important because each position holds a specific value that contributes to different operations such as finding determinants.

When solving for the determinant, as shown in the exercise, it is crucial to carefully identify each matrix element so you can accurately apply the determinant formula.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, matrices, and systems of linear equations. It has a strong application focus in areas such as engineering, physics, computer science, and more.

Key concepts in linear algebra include:

• Vectors: These are quantities that have both magnitude and direction. Vectors can be represented in matrix form, which allows for calculations involving multiple dimensions.
• Matrices: A matrix is a rectangular array of numbers arranged in rows and columns. It is used to represent systems of linear equations and transformations.
• Determinants: The determinant is a special number that can be calculated from a square matrix. It helps in understanding properties of the matrix, such as invertibility.

For instance, the determinant we calculated in this exercise informs us about the matrix's behavior under various transformations and whether it has an inverse or not.

Matrix Operations

Matrix operations are key to manipulating and understanding matrices in linear algebra.

Common operations include:

• Addition and Subtraction: Matrices can be added or subtracted only if they have the same dimensions, i.e., the same number of rows and columns.
• Multiplication: There are different types of multiplication, including scalar multiplication, where each element of the matrix is multiplied by a given scalar, and matrix multiplication, which combines two matrices.
• Determinant Calculation: This operation involves evaluating a square matrix (like our 2x2 matrix) to discover a single number representing the matrix's characteristic. For a 2x2 matrix, the formula is \(ad - bc\) where a, b, c, and d are elements of the matrix.
• Inversion: A matrix is invertible if it has a non-zero determinant. Finding the inverse involves several operations that depend on determinants.

These operations allow us to perform a variety of tasks, from solving systems of equations to transforming geometric data.