Problem 11 - Algebra and Trigonometry

Question

Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{cc}{\frac{1}{2}} & {\frac{1}{8}} \\ {1} & {\frac{1}{2}}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is \( \frac{1}{8} \).

Step 1: Identify the matrix format

The given matrix is a 2x2 matrix: \[\begin{bmatrix} \frac{1}{2} & \frac{1}{8} \1 & \frac{1}{2} \end{bmatrix} \]. This format allows us to use the formula for the determinant of a 2x2 matrix.

Step 2: Recall the determinant formula for 2x2 matrix

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

Step 3: Assign each element to variables

Identify the elements of the matrix such that \(a = \frac{1}{2}\), \(b = \frac{1}{8}\), \(c = 1\), and \(d = \frac{1}{2}\).

Step 4: Substitute the values into the determinant formula

Plug in the values from the matrix into the formula: \[\det(A) = \left( \frac{1}{2} \times \frac{1}{2} \right) - \left( \frac{1}{8} \times 1 \right)\].

Step 5: Calculate the determinant

Perform the calculations: \[\det(A) = \frac{1}{4} - \frac{1}{8}\]. Convert \(\frac{1}{4}\) to \(\frac{2}{8}\) to subtract fractions easily. Thus, \[\det(A) = \frac{2}{8} - \frac{1}{8} = \frac{1}{8}\].

Key Concepts

2x2 matrix

A 2x2 matrix is a simple type of matrix that has two rows and two columns. In the context of matrices, the term '2x2' refers to the size, with the first digit indicating the number of rows and the second digit indicating the number of columns. For example, the matrix \[\begin{bmatrix} \frac{1}{2} & \frac{1}{8} \1 & \frac{1}{2} \end{bmatrix}\] is a good representation of a 2x2 matrix. Understanding the layout and components of a 2x2 matrix is essential. Each element in the matrix belongs to a specific position. This positioning is crucial, especially when performing operations such as determinant calculation, addition, and multiplication. Often, matrices are represented using letters such as A, B, or C, and their components (elements) are often denoted by lowercase letters like \(a, b, c,\) and \(d.\) In this format, a 2x2 matrix is typically denoted as: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]where \(a, b, c, \) and \(d\) are elements of the matrix.

matrix operations

Matrix operations encompass a variety of calculations that can be performed on matrices. With a 2x2 matrix, such operations include addition, subtraction, multiplication, and the calculation of the determinant, as highlighted in the original exercise.

• Addition/Subtraction: To add or subtract matrices, simply add or subtract their corresponding elements. It is important to note that these operations are only possible when the matrices are of the same size.


• Multiplication: This is slightly more complex. The elements of the resulting matrix are determined by multiplying elements across rows and columns and summing the results.


• Determinant: This is a key matrix operation for square matrices like 2x2 matrices. The determinant provides specific information about the matrix, such as whether it has an inverse.

When considering matrix operations, understanding the rules is vital. For instance, while addition and subtraction are straightforward, multiplication follows specific rules of each element interacting with others in a particular manner.

determinant formula

The determinant of a 2x2 matrix is a special number that can be calculated using a straightforward formula. For a matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant, often denoted as \( \,\det(A) \,\), is found using the formula:\[\det(A) = ad - bc\]This formula tells us to multiply the top-left and bottom-right elements (\(a\) and \(d\)) and subtract the product of the top-right and bottom-left elements (\(b\) and \(c\)). In simpler terms, the process involves two multiplications and a subtraction.

• Calculate \(a \times d\)
• Calculate \(b \times c\)
• Subtract: \( (a \times d) - (b \times c) \)

Let's see how this applies to our example matrix. By substituting \(a = \frac{1}{2}, b = \frac{1}{8}, c = 1,\) and \(d = \frac{1}{2}\), the determinant becomes:\[\det(A) = \left(\frac{1}{2} \times \frac{1}{2}\right) - \left(\frac{1}{8} \times 1\right)\]which simplifies to \(\frac{1}{8}\). This determinant indicates that the matrix has a unique solution when applied to systems of linear equations.