Problem 2 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{ll} 7 & 6 \\ 2 & 5 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 23.

Step 1: Understanding the Determinant of a 2x2 Matrix

A 2x2 determinant has the following form: \(\left| \begin{array}{ll} a & b \ c & d \end{array} \right|\). The determinant of this matrix is calculated using the formula: \(ad - bc\). This step helps us understand that we need to identify the elements \(a, b, c,\) and \(d\) within the given matrix.

Step 2: Identifying Matrix Elements

From the given matrix \(\left| \begin{array}{ll} 7 & 6 \ 2 & 5 \end{array} \right|\), assign the numbers: \(a = 7\), \(b = 6\), \(c = 2\), and \(d = 5\). This setup is essential for substituting these values into the determinant formula.

Step 3: Using the Determinant Formula

Substitute the identified values into the determinant formula: \(ad - bc\). Substitute \(a = 7\), \(b = 6\), \(c = 2\), and \(d = 5\) into the formula to compute: \(7 \times 5 - 6 \times 2\).

Step 4: Calculating the Determinant

Perform the multiplication: \(7 \times 5 = 35\) and \(6 \times 2 = 12\). Then, subtract the two results to get: \(35 - 12\).

Step 5: Final Result

Calculate \(35 - 12 = 23\). Therefore, the determinant of the given matrix is 23.

Key Concepts

Understanding 2x2 Matrices

A 2x2 matrix is a simple representation of data in a square format composed of exactly two rows and two columns. It is one of the simplest types of matrices, yet it holds fundamental importance in linear algebra. A matrix is usually denoted by a capital letter, like \( \mathbf{A} \). For example, consider the matrix:\[\mathbf{A} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]Each element in the matrix has a specific position, identified by its row and column number. Understanding how to manipulate and compute characteristics of a 2x2 matrix, like its determinant, is crucial in various applications in math and science. Knowing how such matrices work aids in data transformation and solving linear equations among other tasks.

Matrix Elements and Their Identification

Matrix elements are the individual numerical entries in a matrix. For a 2x2 matrix like our example \(\mathbf{A}\), these elements are denoted as \(a, b, c, \) and \(d\):- The first row contains the elements \(a\) and \(b\).- The second row contains the elements \(c\) and \(d\).For instance, in the matrix:\[\begin{bmatrix} 7 & 6 \ 2 & 5 \end{bmatrix}\]Each number represents an element of the matrix:

• \(a = 7\)
• \(b = 6\)
• \(c = 2\)
• \(d = 5\)

Recognizing and correctly identifying these elements is foundational because they are used in further calculations like determining the determinant, among other operations.

The Determinant Formula for 2x2 Matrices

The determinant of a matrix provides a scalar value that can give us insight into the properties of the matrix, including whether it is invertible. For a 2x2 matrix, calculating the determinant is straightforward using a specific formula: \[det(\mathbf{A}) = ad - bc\]In this formula:

• Multiply the first diagonal's elements, \(a\) and \(d\), to get \(ad\).
• Multiply the second diagonal's elements, \(b\) and \(c\), to get \(bc\).
• Subtract the second product from the first: \( ad - bc \).

For our matrix \(\begin{bmatrix} 7 & 6 \ 2 & 5 \end{bmatrix}\):- Calculate \(7 \times 5 = 35\).- Calculate \(6 \times 2 = 12\).- Subtract \(12\) from \(35\), getting a determinant of \(23\).This final scalar value, \(23\), tells us about the matrix's area scaling factor and properties such as invertibility. If a 2x2 matrix has a determinant of zero, it means the matrix does not have an inverse, limiting certain functionalities.