Problem 2 - Algebra and Trigonometry

Question

Evaluate each determinant. $$ \left|\begin{array}{ll} {4} & {8} \\ {5} & {6} \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is -16.

Step 1: Identify the main diagonal

The main diagonal of a 2x2 matrix runs from the top left to the bottom right, so the numbers on the main diagonal are 4 and 6.

Step 2: Identify the 'off' diagonal

The 'off' diagonal of a 2x2 matrix runs from the top right to the bottom left, so the numbers on the 'off' diagonal are 8 and 5.

Step 3: Calculate the determinant

The determinant of a 2x2 matrix is calculated by multiplying the elements on the main diagonal, and subtracting the product of the elements off the main diagonal. So, \((4 \times 6) - (8 \times 5) = 24 - 40 = -16\).

Key Concepts

Matrix Algebra

Matrix algebra is a fundamental part of mathematics that deals with matrices, rows, and columns. A matrix is a rectangular array of numbers organized into rows and columns, which can represent various data structures or mathematical operations.

In a 2x2 matrix, which we often encounter in introductory courses, there are two rows and two columns, making up a total of four elements. Understanding how to manipulate and compute things like determinants involves recognizing how the rows and columns interact.

• Row: The horizontal set of numbers in a matrix.
• Column: The vertical set of numbers in a matrix.
• Element: The individual numbers inside a matrix.

The importance of matrix algebra lies in its ability to solve linear equations, transform geometric figures, and represent linear transformations.

Evaluating Determinants

The determinant of a 2x2 matrix provides crucial information about the matrix, including whether it has an inverse and the area of the parallelogram it spans. Evaluating the determinant of a 2x2 matrix is a simple process using the formula:

For the matrix: \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
The determinant is calculated as: \[ad - bc\]

• Main Diagonal: Runs from the top left to the bottom right, contributing to the product \(a \times d\).
• Off Diagonal: Runs from the top right to the bottom left, contributing to the product \(b \times c\).
• Final Step: Subtract the off diagonal product from the main diagonal product.

This method simplifies checking matrix properties and solving related problems easily.

Step-by-Step Problem Solving

Solving mathematical problems, especially in matrix algebra, is less daunting when broken into methodical steps. Let's go through the straightforward process to solve a 2x2 determinant problem.

• Step 1: Identify the Main Diagonal
  Find the elements that form the main diagonal. For the given matrix \(\begin{bmatrix}4 & 8 \ 5 & 6\end{bmatrix}\), the main diagonal is 4 and 6.
• Step 2: Identify the Off Diagonal
  Next, identify the elements in the off diagonal, which are 8 and 5.
• Step 3: Compute the Determinant
  Apply the formula: Multiply the main diagonal elements, \(4 \times 6 = 24\), and the off diagonal elements, \(8 \times 5 = 40\).
  Finally, subtract: \(24 - 40 = -16\).

This approach ensures clarity in the problem-solving process and aids in understanding each step along the way.