Problem 6 - A Graphical Approach to College Algebra

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}0 & 2 \\\1 & 5\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -2.

Step 1: Understand the Determinant Formula for a 2x2 Matrix

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated using the formula: \( ad - bc \). This means we multiply the elements on the main diagonal and subtract the product of the elements on the off diagonal.

Step 2: Identify Elements of the Matrix

Identify the elements of the given matrix \( \begin{bmatrix} 0 & 2 \ 1 & 5 \end{bmatrix} \). Here, \(a = 0\), \(b = 2\), \(c = 1\), and \(d = 5\).

Step 3: Apply the Determinant Formula

Substitute the identified values into the determinant formula: \[ \det(A) = (0)(5) - (2)(1) \].

Step 4: Calculate the Determinant

Perform the arithmetic operations: \[ \det(A) = 0 - 2 = -2 \]. This is the result after simplifying the expression by performing multiplication first, followed by subtraction.

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a fundamental concept in linear algebra and consists of two rows and two columns. Structurally, it can be represented as follows:

• The top row consists of elements \(a\) and \(b\).
• The bottom row consists of elements \(c\) and \(d\).

A typical 2x2 matrix looks like this: \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\).
This basic form allows you to perform various operations such as addition, subtraction, and finding the determinant. The beauty of a 2x2 matrix is its simplicity while still being extremely useful in practical problems. Once we grasp the structure, it becomes easier to move on to more complex matrix operations.

Basics of Matrix Arithmetic

Matrix arithmetic includes operations like addition, subtraction, and multiplication. While 2x2 matrices are simple, the operations can become tricky if not done with care. Here’s a quick run-through:

• Addition: Add corresponding elements from the two matrices. For example, if \(\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}\), the resulting matrix is \(\begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}\).
• Subtraction: Similar to addition, subtract each element of one matrix from the corresponding element of the other.
• Multiplication: Although less intuitive, it's crucial. For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and another matrix \( B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\), the product \( AB \) is calculated by multiplying rows by columns which results in another 2x2 matrix.

Understanding these operations will aid in effectively handling and solving matrix-related problems.

Determinant Formula and Its Application

The determinant formula for a 2x2 matrix is a straightforward calculation, essential for many mathematical applications like solving systems of equations or finding matrix inverses. Given a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as:\[det(A) = ad - bc\]This involves finding the product of the elements on the main diagonal (\(a\) and \(d\)) and subtracting the product of the elements on the anti-diagonal (\(b\) and \(c\)).
For example, for the matrix \(\begin{bmatrix} 0 & 2 \ 1 & 5 \end{bmatrix}\), we first identify elements as \(a = 0\), \(b = 2\), \(c = 1\), and \(d = 5\). Plugging these into the formula gives us \[det(A) = (0 \times 5) - (2 \times 1) = 0 - 2 = -2\].This result helps us in understanding properties such as whether a matrix is invertible or not.