Problem 81 - Algebra 2

Question

Use a graphing calculator to find the value of each determinant. $$ \left|\begin{array}{ll}{6.1} & {4.8} \\ {9.7} & {3.5}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -25.21.

Step 1: Understanding the Determinant Formula

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula \( ad - bc \). In this problem, we will identify \(a = 6.1\), \(b = 4.8\), \(c = 9.7\), and \(d = 3.5\).

Step 2: Apply the Determinant Formula

Plug the values into the formula: \( 6.1 \times 3.5 - 4.8 \times 9.7 \). First, calculate \( 6.1 \times 3.5 \) and \( 4.8 \times 9.7 \).

Step 3: Calculate Each Product

Calculate \( 6.1 \times 3.5 = 21.35 \) and \( 4.8 \times 9.7 = 46.56 \).

Step 4: Subtract the Products

Finally, subtract these two results to find the determinant: \( 21.35 - 46.56 = -25.21 \).

Step 5: Verify with Graphing Calculator

Use a graphing calculator to verify the calculation of the determinant \( -25.21 \). Input the matrix values into the calculator and confirm the result matches our manual calculation.

Key Concepts

2x2 matrices

A 2x2 matrix is a straightforward mathematical concept often used in linear algebra. It is essentially a grid presenting two rows and two columns. Each of these grids contains a number, forming a set of four numbers. These matrices are compact and manageable, making them excellent tools for various linear calculations.
To better understand a 2x2 matrix, consider the format:

• Row 1: First number is from the top-left corner, often labeled 'a', and the second number is the top-right corner labeled 'b'.
• Row 2: This row contains 'c' at the bottom-left and 'd' at the bottom-right.

Referring to them with the standard notation, we often denote a 2x2 matrix as:

• \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]

Interacting with these matrices, we conduct operations such as addition, multiplication, and finding determinants, making them vital in mathematics.

matrix multiplication

Matrix multiplication involves combining rows and columns from different matrices. It is an essential mathematical operation, but it can initially seem complex due to the precise nature of how it's performed. Here's how it works for two 2x2 matrices:
In matrix multiplication, you multiply the rows of the first matrix by the columns of the second matrix, following these steps:

• Multiply each element of the first row in the first matrix by the corresponding element in the first column of the second matrix.
• Continue this by following through both rows and columns, ensuring you calculate each position in the resulting matrix correctly.

Thus, if you have two matrices:

• Matrix A: \[\begin{pmatrix} a_1 & b_1 \ c_1 & d_1 \end{pmatrix}\]
• Matrix B: \[\begin{pmatrix} a_2 & b_2 \ c_2 & d_2 \end{pmatrix}\]

The resulting matrix after multiplication will be:

• \[\begin{pmatrix} (a_1\cdot a_2 + b_1\cdot c_2) & (a_1\cdot b_2 + b_1\cdot d_2) \ (c_1\cdot a_2 + d_1\cdot c_2) & (c_1\cdot b_2 + d_1\cdot d_2) \end{pmatrix}\]

This operation helps in transforming vectors and matrices in various applications such as computer graphics and solving linear equations.

graphing calculator

A graphing calculator is a handy tool, especially when dealing with matrices and their operations. It is capable of not only calculating determinants quickly but also solving equations visually and numerically.
To use a graphing calculator for determining a 2x2 matrix, follow these simple steps:

• Access the matrix function available on your calculator, usually under a menu labeled with "matrix" or "mat".
• Select the 'edit' function to input the matrix size and values, which, for a 2x2 matrix, means setting both the row and column to 2.
• Input your specific values, ensuring that they are placed correctly in the format as mentioned above.
• Use the determinant function, often found in the same matrix menu, to calculate the determinant.

The graphing calculator will quickly deliver the determinant result, allowing you to verify any manual calculations you've performed. It eradicates tedious calculations and paves the way for focusing on analyzing results.

mathematical verification

Mathematical verification is a process by which we confirm the accuracy of our calculations. It is critical in ensuring that the found solution to a problem holds true. With matrices, this is especially vital due to the potential for simple arithmetic errors leading to drastically different results.
To verify a determinant calculated by hand, you could follow these steps:

• Re-examine each step of your manual calculation, checking each multiplication and subtraction.
• Use alternative methods, such as a graphing calculator, to double-check the output.
• If discrepancies occur, trace the mistake back to its origin, whether it lies in the multiplication, subtraction, or input of values.

By utilizing tools and cross-verifying, you ensure that the results are consistent and accurate. Verification not only validates your problem-solving techniques but also instills confidence in your mathematical ability. Whether manually or with technology, having this layer of assurance is incredibly beneficial for any math-related task.