Problem 26 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{cc} 20 & -3 \\ 20 & -3 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Recall the Formula for 2x2 Determinant

The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is calculated using the formula \(ad - bc\).

Step 2: Identify the Elements

From the matrix \(\begin{pmatrix} 20 & -3 \ 20 & -3 \end{pmatrix}\), identify \(a = 20\), \(b = -3\), \(c = 20\), and \(d = -3\).

Step 3: Plug the Values into the Formula

Substitute the values into the determinant formula: \[ ext{Determinant} = (20)(-3) - (20)(-3) \]

Step 4: Perform the Calculations

Calculate each term: \((20)(-3) = -60\) and \((20)(-3) = -60\).Subtract these results: \[-60 - (-60) = -60 + 60 = 0\].

Step 5: Final Answer

The determinant of the matrix is 0. Since the rows are identical, it results in zero, which is a known property of determinants with identical rows or columns.

Key Concepts

Understanding 2x2 Matrices

When studying linear algebra, one of the basic structures you will encounter is the 2x2 matrix. These matrices consist of two rows and two columns, creating a grid of four elements. Imagine a small table with four slots where numbers sit like guests at a dinner party. Each element in the matrix can have its own role and contributes to the calculations you perform using the matrix. Mathematically, you might see a 2x2 matrix written like this:
\[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]In this grid, \(a\), \(b\), \(c\), and \(d\) represent numbers or variables that you will use in your calculations. Understanding the layout helps you see how the matrix operates in greater mathematical contexts. You use these matrices for tasks ranging from solving systems of equations to transformations in graphics.

Key Properties of Matrices

Matrices have properties that are crucial for understanding their behavior in math. For instance, a notable property is how the determinant can reveal whether a matrix can be inverted or its effect on a space it transforms. Particularly, the determinant of a matrix is a single number derived from its elements that provides important information about the matrix.
In the case of 2x2 matrices, a helpful shortcut is knowing that if any two rows (or columns) are identical, the determinant will be zero. This tells us the matrix's transformation action would flatten space or make vectors collapse into a lower dimension, which is key when considering concepts like matrix invertibility.

Exploring Linear Algebra Concepts

Linear algebra is all about working with vectors and matrices to understand transformations and solve equations. Within this field, you often use determinants to understand how matrices influence space. They give you insight into whether transformations distort space or preserve volume.
When you look at a 2x2 matrix, calculating its determinant helps determine whether it's invertible. Invertibility implies you can 'undo' the transformation that a matrix applies. Specifically, if a determinant equals zero, the matrix lacks an inverse due to its flattening characteristic. This insight is not just theoretical but has practical applications in computer graphics, physics simulations, and other real-world problem-solving scenarios. Learning these concepts provides a solid foundation for more advanced studies in mathematics and its applications.