Problem 18 - Precalculus with Limits

Question

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} 0 && 6 \\ -3 && 2 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the matrix is 18.

Step 1: Identify the elements in the diagonal and auxiliary diagonal

In our matrix, the elements in the main diagonal are 0 and 2. The elements in the auxiliary diagonal are 6 and -3.

Step 2: Apply the determinant formula

We multiply the elements in the main diagonal and do the same for the auxiliary diagonal. Then we subtract the second result from the first one. This gives us: \( (0*2) - (6*-3) \)

Step 3: Solve the calculation

After performing the multiplication and subtraction, we get:\( (0) - (-18) = 18 \)

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple rectangular array with two rows and two columns. It is one of the most basic structures in linear algebra. You can think of each element as occupying a cell within this 2x2 grid. For any 2x2 matrix:\[\begin{pmatrix}a && b \c && d \end{pmatrix}\]Each letter represents a number or variable placed in a specific position.

• Here, "a" and "b" form the first row.
• "c" and "d" form the second row.

Understanding how these numbers interact is the first step towards solving many algebraic problems including finding the determinant. The determinant tells us important properties about the matrix, such as whether it is invertible.

Main Diagonal

The main diagonal of a matrix encompasses the elements stretching from the top-left to the bottom-right. For a 2x2 matrix, it includes just two elements. Consider the matrix:\[\begin{pmatrix}0 && 6 \-3 && 2 \end{pmatrix}\]In this specific case, the main diagonal elements are 0 and 2.

• These elements are crucial for several matrix operations.
• They affect how we calculate the determinant of the matrix.

Recognizing the main diagonal helps you know which elements to multiply initially for determinant calculations.

Auxiliary Diagonal

The auxiliary diagonal in a matrix runs opposite to the main diagonal, from the top-right to the bottom-left. In our matrix:\[\begin{pmatrix}0 && 6 \-3 && 2 \end{pmatrix}\]The auxiliary diagonal consists of the elements 6 and -3.

• It's essential for determining the cross-product you need for calculating the determinant.
• Understanding this helps complete the second half of the determinant formula.

Recognizing both diagonals ensures a correct approach in step-by-step calculations.

Determinant Calculation

Calculating the determinant of a 2x2 matrix is straightforward once you identify the elements in both the main and auxiliary diagonals. The formula for the determinant of a 2x2 matrix is given by:\[ ext{Det} = (a \cdot d) - (b \cdot c)\]Applying this to our matrix, where "a" is 0, "b" is 6, "c" is -3, and "d" is 2, we perform the calculation:\[(0 \cdot 2) - (6 \cdot -3) = 0 - (-18) = 18\]

• The multiplication of the main diagonal elements gives 0.
• The multiplication of the auxiliary diagonal elements gives -18.
• The subtraction of these results in a determinant of 18.

This simple operation not only aids in understanding specific matrix properties but also exemplifies the elegance of linear algebra.