Problem 90 - Precalculus

Question

Explain the mistake that is made. $$\text { Multiply }\left[\begin{array}{ll}3 & 2 \\\1 & 4\end{array}\right]\left[\begin{array}{ll}-1 & 3 \\\\-2 & 5\end{array}\right]$$ Solution: Multiply using column-by-row method. \(\left[\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right]\left[\begin{array}{ll}-1 & 3 \\ -2 & 5\end{array}\right]=\left[\begin{array}{ll}(3)(-1)+(1)(3) & (2)(-1)+(4)(3) \\\ (3)(-2)+(1)(5) & (2)(-2)+(4)(5)\end{array}\right]\) Simplify. \(\quad\left[\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right]\left[\begin{array}{ll}-1 & 3 \\ -2 & 5\end{array}\right]=\left[\begin{array}{rr}0 & 10 \\ -1 & 16\end{array}\right]\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was incorrect calculations of the product's elements, resulting in wrong summations.

Step 1: Setup the multiplication

Set up the matrix multiplication problem: Multiply two 2x2 matrices as follows:Given matrices: \[A = \begin{bmatrix} 3 & 2 \ 1 & 4 \end{bmatrix},\ B = \begin{bmatrix} -1 & 3 \ -2 & 5 \end{bmatrix}\]The product, \( AB \), is calculated by taking the dot product of rows from \( A \) with columns from \( B \).

Step 2: Calculate the first row of the product

For the first row of the product matrix, multiply the rows of \( A \) by the columns of \( B \):1. First element (1st row, 1st column):\( 3 \times (-1) + 2 \times (-2) = -3 - 4 = -7 \)2. Second element (1st row, 2nd column):\( 3 \times 3 + 2 \times 5 = 9 + 10 = 19 \)Thus, the first row of the resulting matrix is \([ -7, 19 ]\).

Step 3: Calculate the second row of the product

For the second row of the product matrix, continue using the same method:1. Third element (2nd row, 1st column):\( 1 \times (-1) + 4 \times (-2) = -1 - 8 = -9 \)2. Fourth element (2nd row, 2nd column):\( 1 \times 3 + 4 \times 5 = 3 + 20 = 23 \)Thus, the second row of the resulting matrix is \([ -9, 23 ]\).

Step 4: Write down the correct product matrix

Combine the rows to form the full resulting matrix:\[\begin{bmatrix} -7 & 19 \ -9 & 23 \end{bmatrix}\]This is the correct result for the matrix multiplication.

Step 5: Identify the mistake in the solution

The mistake in the provided solution was in the calculation of each element of the resulting matrix. The student used incorrect multipliers from the matrices and made arithmetic errors. For example, the first entry was calculated as \(0\) instead of \(-7\). Always ensure to multiply corresponding elements correctly and sum them accurately.

Key Concepts

2x2 Matrices

Matrix multiplication is an essential concept in linear algebra and comes into play in various fields like physics, computer science, and engineering. The matrices involved in the exercise here are 2x2 matrices, which means they have 2 rows and 2 columns each. When multiplying two 2x2 matrices, the product is also a 2x2 matrix.

Each element in the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. This process involves multiple simple arithmetic operations. Let's break down how you can intuitively remember this for a 2x2 matrix:

• For the first row, first column of the product, multiply the first row of the first matrix by the first column of the second matrix.
• For the first row, second column, multiply the first row of the first matrix by the second column of the second matrix.
• For the second row, first column, multiply the second row of the first matrix by the first column of the second matrix.
• For the second row, second column, multiply the second row of the first matrix by the second column of the second matrix.

Understanding and remembering the correct setup allows you to approach each multiplication correctly, reducing errors in calculations.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation when multiplying matrices. In the context of our problem with 2x2 matrices, it involves multiplying each element from the row of the first matrix with the corresponding element from the column of the second matrix, and then summing these products.

The dot product helps us to determine each element of the resulting matrix. Consider the first element in our example:

• Take the first row from the first matrix: [3, 2].
• Take the first column from the second matrix: [-1, -2].
• Multiply and sum: - The first multiplication is 3 times -1, which equals -3.
  - The second multiplication is 2 times -2, which equals -4.
  - Adding these, we get -7.

This resultant value is the element in the first row, first column of the product matrix.

While the calculation is straightforward, it is crucial to maintain accuracy, especially when moving systematically through the matrix. By mastering the dot product technique, you can effectively multiply matrices of any size.

Arithmetic Errors

Arithmetic errors can occur during matrix multiplication if calculations are rushed or inattentively performed. These errors often stem from incorrect multipliers or addition mistakes, which can lead to inaccurate results. In the exercise provided, the student made a few arithmetic mistakes, which affected the final outcome significantly.

Common arithmetic errors include:

• Multiplying incorrect pairs of numbers, leading to wrong intermediate results.
• Adding results incorrectly, which can result in a wrong final entry for the resulting matrix.
• Skipping numbers or steps, or swapping them by mistake, which can disrupt the entire calculation.

To minimize these errors:

• Always double-check your multipliers to ensure they are correct.
• Carefully sum the products to avoid mistakes.
• Break down calculations into smaller, manageable pieces if necessary, rather than trying to solve large problems in one step.
• Practice regularly, as familiarity with the process will boost both confidence and accuracy.

A consistent and methodical approach, paired with careful checking, will significantly reduce the likelihood of arithmetic errors in matrix multiplication.