Problem 7 - College Algebra

Question

For the following exercises, find the determinant. \(\left|\begin{array}{rr}2 & -5 \\ -1 & 6\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant of the matrix is 7.

Step 1: Identify the matrix

The given matrix is a 2x2 matrix: \[ A = \begin{pmatrix} 2 & -5 \ -1 & 6 \end{pmatrix} \]. A 2x2 matrix has the form \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) where \( a=2 \), \( b=-5 \), \( c=-1 \), and \( d=6 \).

Step 2: Apply the determinant formula for a 2x2 matrix

The formula to find the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \( ad - bc \). For matrix \( A \), substitute the values: \( a=2 \), \( b=-5 \), \( c=-1 \), and \( d=6 \).

Step 3: Calculate the products

Calculate the product \( ad \), which is \( 2 \times 6 = 12 \), and the product \( bc \), which is \( -5 \times -1 = 5 \).

Step 4: Compute the determinant

Subtract the second product from the first: \( 12 - 5 = 7 \). Thus, the determinant of the matrix is 7.

Key Concepts

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, having only two rows and two columns. This makes it a square matrix that is relatively easy to work with. Each entry in this type of matrix can be identified with a specific position. In general, a 2x2 matrix looks like this: \[\begin{pmatrix} a & b \ c & d \end{pmatrix} \]where:

• \(a\) is the element in the first row, first column
• \(b\) is the element in the first row, second column
• \(c\) is the element in the second row, first column
• \(d\) is the element in the second row, second column

The simplicity of a 2x2 matrix is advantageous as it allows us to easily perform calculations like finding determinants, which are crucial in various mathematical applications.

Matrix Operations

Matrix operations involve various calculations and transformations that can be applied to matrices, such as addition, subtraction, multiplication, and finding determinants. In this context, we are focusing on the operation of finding the determinant of a 2x2 matrix. For a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated using the formula \( ad - bc \). This is a straightforward operation where you:

• Multiply the elements on the main diagonal, \(a\) and \(d\), which is the product \(ad\).
• Multiply the elements on the counter-diagonal, \(b\) and \(c\), which gives \(bc\).
• Subtract the counter-diagonal product from the main diagonal product, yielding \(ad - bc\).

Understanding matrix operations, especially how to compute determinants, is fundamental in linear algebra, as these calculations are used in systems of equations and other advanced concepts.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and linear transformations. It plays a critical role in fields such as engineering, physics, computer science, and economics. Central to linear algebra is the concept of matrices and determinants. Determinants in particular are crucial because they help us assess matrix properties, like invertibility. Specifically, a square matrix (including 2x2 matrices) is invertible if its determinant is not zero. Here's why it matters:

• An invertible matrix corresponds to a linear transformation that is bijective, meaning it has a unique inverse operation.
• In systems of linear equations, determinants help determine whether a unique solution exists.

The determinant of a matrix is indicative of the scaling factor applied by the corresponding linear transformation. A zero determinant indicates that the transformation maps to a lower dimension, often implying a loss of information or a lack of unique solutions.