Problem 4 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} -2 & 1 \\ 3 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 1.

Step 1: Understand the Matrix

The given matrix is a 2x2 square matrix:\[\begin{bmatrix} -2 & 1 \ 3 & -2 \end{bmatrix}\]Understanding it's a 2x2 matrix, confirms that we can find its determinant using the formula for 2x2 matrices.

Step 2: Recall the Determinant Formula

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\). We'll apply this formula to our matrix.

Step 3: Identify the Elements

In our matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} -2 & 1 \ 3 & -2 \end{bmatrix}\]we have: \(a = -2\), \(b = 1\), \(c = 3\), \(d = -2\).

Step 4: Substitute into the Formula

Substitute the values into the determinant formula:\[det(A) = (-2)(-2) - (1)(3)\]which simplifies to \(4 - 3\).

Step 5: Calculate the Determinant

Perform the subtraction:\[det(A) = 4 - 3 = 1\]Thus, the determinant of the matrix is 1.

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a special type of mathematical object arranged in a square grid with 2 rows and 2 columns. Each element in the matrix is defined by its position within these rows and columns. For instance, in our example matrix \(\begin{bmatrix} -2 & 1 \ 3 & -2 \end{bmatrix}\), we organize the numbers in the following layout: the top row contains \(-2\) and \(1\), while the bottom row consists of \(3\) and \(-2\).
The layout gives us a compact way to represent numbers that can apply to real-life problems such as systems of equations, transformations, or data organization. Understanding the layout makes further operations on matrices, like calculating determinants, straightforward.

• The matrix has two rows: \(\begin{bmatrix} -2 & 1 \end{bmatrix}\) and \(\begin{bmatrix} 3 & -2 \end{bmatrix}\).
• The matrix has two columns: \(\begin{bmatrix} -2 \ 3 \end{bmatrix}\) and \(\begin{bmatrix} 1 \ -2 \end{bmatrix}\).
• Elements of the matrix are represented as \(a, b, c, \) and \(d\) for calculation purposes.

The Determinant Formula for a 2x2 Matrix

To calculate the determinant of a 2x2 matrix, you use a straightforward formula. This formula is essential in understanding concepts like inverses of matrices or changes in linear transformations. The formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given as:
\[\det(A) = ad - bc\]This expression cleverly combines the diagonal elements of the matrix:

• Multiply the first diagonal (from top-left to bottom-right) so \(a \, \times \, d\).
• Multiply the second diagonal (from top-right to bottom-left) so \(b \, \times \, c\).
• Subtract the result of the second diagonal from the first: \(ad - bc\).

This formula is unique to 2x2 matrices, simplifying the calculation process compared to larger matrices.

Performing the Matrix Calculation

With our given matrix \(\begin{bmatrix} -2 & 1 \ 3 & -2 \end{bmatrix}\), we proceed to determine its determinant using the formula \(ad - bc\).The first step is identifying the values of \(a, b, c, \) and \(d\):

• Here, \(a = -2\), \(b = 1\), \(c = 3\), and \(d = -2\).

Next, we substitute these values into the determinant formula:
\[\det(A) = (-2) \cdot (-2) - (1) \cdot (3)\]
This calculation breaks down into two main multiplication steps:

• Calculate the first product: \((-2) \cdot (-2) = 4\).
• Calculate the second product: \(1 \cdot 3 = 3\).

Finally, subtract the second product from the first. So,\[\det(A) = 4 - 3 = 1\]Therefore, the determinant of the matrix is \(1\), concluding our calculation.