Problem 38 - Algebra 2

Question

FIND THE ERROR Khalid and Erica are finding the determinant of \(\left[\begin{array}{cc}{8} & {3} \\ {-5} & {2}\end{array}\right]\) Who is correct? Explain your reasoning. $$ \begin{array}{c}{\text { Khalid }} \\ {\left|\begin{array}{rr}{8} & {3} \\\ {-5} & {2}\end{array}\right|=16-(-15)} \\ {=31}\end{array} $$ $$ \begin{aligned} & \text {Erica} \\\\\left|\begin{array}{cc}{8} & {3} \\ {-5} & {2}\end{array}\right| &=16-15 \\ &=1 \end{aligned} $$

Step-by-Step Solution

Answer

Khalid is correct; the determinant is 31.

Step 1: Understand the Determinant Formula for 2x2 Matrices

The determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is calculated using the formula \( ad - bc \). Let's use this formula to find the determinant of the given matrix.

Step 2: Apply the Determinant Formula to the Matrix

The given matrix is \( \left[ \begin{array}{cc} 8 & 3 \ -5 & 2 \end{array} \right] \). According to the formula, calculate the determinant as follows: \( (8)(2) - (3)(-5) = 16 - (-15) \).

Step 3: Simplify the Calculation

Simplify \( 16 - (-15) \) by recognizing that subtracting a negative is the same as adding: \( 16 + 15 = 31 \).

Step 4: Identify Errors in Khalid and Erica's Solutions

Khalid correctly calculated: \( 16 - (-15) = 31 \). Erica incorrectly calculated: \( 16 - 15 = 1 \), which doesn’t apply here.

Step 5: Conclusion

Khalid is correct. The error in Erica's solution came from not properly handling the subtraction of a negative number.

Key Concepts

2x2 Matrices

2x2 matrices are a fundamental concept in linear algebra. These matrices consist of two rows and two columns. You can think of a 2x2 matrix as a small table of numbers arranged in a square. Each number in the matrix has a specific position, and these positions are important when performing matrix operations.
For example, consider the matrix \[ \left[ \begin{array}{cc} 8 & 3 \ -5 & 2 \end{array} \right] \].
Here, '8' is at the top left, and '3' is next to it on the top right. Similarly, '-5' is on the bottom left, and '2' is on the bottom right. Understanding the layout helps when applying operations like finding determinants or multiplying matrices.

Matrix Operations

Matrix operations involve various mathematical procedures applied to matrices, such as addition, subtraction, and finding determinants. These operations follow specific rules to ensure consistency and accuracy.

• Addition and Subtraction: You can only add or subtract matrices of the same size. Simply add or subtract corresponding elements.
• Multiplication: Matrix multiplication has unique rules. For example, in multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second.
• Determinant Calculation: For a 2x2 matrix, the determinant is calculated as \( ad - bc \), where \( a, b, c, \) and \( d \) are the elements of the matrix.

In our example matrix, we calculate the determinant by multiplying and subtracting as per the formula. Check your steps to avoid mistakes.

Mathematical Errors

Mathematical errors can often arise from simple missteps or misunderstandings. In matrix math, these are usually due to incorrect arithmetic or misapplication of operations.
One common error occurs during the subtraction of negative numbers. For instance, in finding determinants, if you have \( 16 - (-15) \), remember that subtracting a negative is equivalent to adding a positive: \( 16 + 15 \).

Here are some tips to avoid such errors:

• Double-check signs: Always revisit your calculations to ensure every positive and negative sign is correctly handled.
• Re-calculate: Perform calculations more than once to confirm accuracy. This is especially important with tricky signs.
• Understand the formula: Know the determinant formula by heart and follow each step carefully.

By applying these strategies, students can minimize errors and improve their problem-solving abilities.