Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-5 & 9 \\\4 & -1\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -31.

Step 1: Identify the Matrix Components

The given matrix is a 2x2 matrix: \(\begin{bmatrix} -5 & 9 \ 4 & -1\end{bmatrix}\). We denote the matrix components as \(A = -5\), \(B = 9\), \(C = 4\), and \(D = -1\).

Step 2: Recall the Determinant Formula for a 2x2 Matrix

For a 2x2 matrix \(\begin{bmatrix} A & B \ C & D\end{bmatrix}\), the determinant \(\text{det}(A)\) is calculated using the formula: \[\text{det}(A) = A \cdot D - B \cdot C\]

Step 3: Substitute the Values into the Formula

Substitute \(A = -5\), \(B = 9\), \(C = 4\), and \(D = -1\) into the determinant formula: \[\text{det}(A) = (-5) \cdot (-1) - 9 \cdot 4\]

Step 4: Perform the Calculations

Calculate each part of the formula: - The product of \(-5\) and \(-1\) is \(5\).- The product of \(9\) and \(4\) is \(36\).Subtract the two results: \[5 - 36 = -31\]

Step 5: Write the Final Answer

The determinant of the matrix is \(-31\).

Key Concepts

2x2 matrix

When you encounter a 2x2 matrix, think of a small grid that has two rows and two columns.
This type of matrix is one of the simplest forms you might come across in mathematics. Each grid square in the matrix holds a number, which we call an element. So, when you read or hear about a "2x2 matrix," it's simply a way to refer to this two-by-two grid.

The syntax for writing a 2x2 matrix looks like this:

• The top left position houses the element often named \( A \).
• The top right position holds \( B \).
• The bottom left position is \( C \).
• The bottom right houses \( D \).

If someone hands you a matrix \( \begin{bmatrix} x & y \ z & w \end{bmatrix} \), they are defining a 2x2 matrix with elements \( x, y, z, \) and \( w \).
Understanding the layout and positioning of these elements is essential for performing operations like finding a determinant.

matrix components

Matrix components are the individual elements that make up a matrix.
In the context of a 2x2 matrix, these are denoted as \(A, B, C,\) and \(D\). It is the combination of these components that allows us to perform mathematical operations.

Understanding the role of each component is crucial:

• \( A \) and \( D \) are on the diagonal running from top left to bottom right.
• \( B \) is the top right, and \( C \) is the bottom left element, forming the off-diagonal.

These components might seem like mere numbers, but they hold vital information for operations like calculating determinants.
In the exercise, we specifically dealt with \( A = -5, B = 9, C = 4, \) and \( D = -1 \). Knowing these details allows us to apply formulas accurately and derive correct results.

matrices operation

Matrices operations take various forms depending on what you want to achieve. Here, we focus specifically on finding the determinant of a 2x2 matrix.
The determinant is a special number that translates aspects of a matrix into a single value, influencing the matrix's properties.

For a 2x2 matrix given as:\[\begin{bmatrix}x & y \z & w\end{bmatrix}\]The determinant \( \text{det}(A) \) is calculated using the formula:\[\text{det}(A) = x \cdot w - y \cdot z\]This formula tells us:

• Multiply the diagonal elements (top left and bottom right).
• Subtract the product of the off-diagonal elements (top right and bottom left).

In our original exercise, by substituting the values in \( \text{det}(A) = -5 \cdot -1 - 9 \cdot 4 \), we executed the operation to find that the determinant is \(-31\).
Mastering this operation unlocks deeper insights into matrix behavior.