Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Matrix Multiplication Use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$A=\left[\begin{array}{rrrr} -3 & 8 & -6 & 8 \\ -12 & 15 & 9 & 6 \\ 5 & -1 & 1 & 5 \end{array}\right], \quad B=\left[\begin{array}{rrr} 3 & 1 & 6 \\ 24 & 15 & 14 \\ 16 & 10 & 21 \\ 8 & -4 & 10 \end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix will be a 3x3 matrix obtained by taking the dot product of each row in matrix \(A\) with the corresponding column in matrix \(B\). Use a graphing utility to perform this operation to get the exact results. But remember, each element in the product matrix is obtained by finding the dot product of the corresponding row from the first matrix and the column from the second matrix.

Step 1: Prepare for matrix multiplication

Check the matrices \(A\) and \(B\) to make sure they can be multiplied together. This is true if the number of columns in the first matrix equals the number of rows in the second matrix. Since matrix\(A\) has 4 columns and matrix \(B\) has 4 rows, matrix multiplication is possible.

Step 2: Set up the result matrix

Set up a new matrix that will hold the results of the multiplication. This matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. So, it will be a 3x3 matrix.

Step 3: Perform the multiplication

Each element in the result matrix is the dot product of the corresponding row in the first matrix and column in the second matrix. For instance, to get element (1,1) in the result matrix, multiply each element of row 1 in matrix \(A\) by the corresponding element of column 1 in matrix \(B\) and sum them up. Perform this operation for all elements in the new matrix.

Key Concepts

Matrix Operations

Matrix operations are a set of procedures you can perform on matrices, which are mathematical objects arranged in a grid format with rows and columns.
These operations include addition, subtraction, and importantly, multiplication.
While addition and subtraction of matrices are straightforward—performed element-wise—matrix multiplication is more complex.
To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second.
This step is crucial as it sets the foundation for matrix multiplication resulting in a new matrix.
Notably, the resulting matrix dimensions reflect the number of rows from the first matrix and the columns from the second, requiring careful preparation on how to proceed.
Let's delve deeper into how this operation leads to the creation of a solution matrix by exploring the dot product method.

Dot Product

The dot product is a pivotal operation used to multiply matrices.
In matrix multiplication, each element of the resulting matrix is determined by calculating the dot product of corresponding rows and columns from the matrices being multiplied.
The process involves multiplying each element from a row of the first matrix by the corresponding element in a column of the second matrix and then summing the results.
Here's a detailed breakdown of the operation:

• Take the first row of the first matrix and pair each element with the corresponding elements from the first column of the second matrix.
• Multiply each pair of numbers together, then sum these products. This sum is the first element of the resulting matrix.
• Repeat the process for each combination of rows and columns needed, eventually filling in every spot in the new matrix.

This systematic way of matrix multiplication highlights the significance of properly aligning matrix dimensions and proficiently performing element-wise multiplications followed by summation.

Graphing Utility

A graphing utility can be an invaluable tool during matrix operations, especially in performing tedious calculations inherent in matrix multiplication.
These tools are often built into graphing calculators or software and are designed to handle complex mathematical operations quickly and accurately.
Using a graphing utility, you can:

• Easily input matrices using a user-friendly interface.
• Automatically perform matrix multiplication without manual computation.
• Visually verify results through graphical interfaces that enhance understanding.

By using such technological aids, students can focus more on understanding process mechanics and less on manual computation errors, solidifying their learning of concepts such as the dot product and matrix dimensionality requirements.

Matrix Dimensions

Understanding matrix dimensions is essential for executing matrix operations correctly.
A matrix's dimensions are given in terms of rows and columns, stated as m x n, where m is the number of rows and n is the number of columns.
For successful matrix multiplication, the key requirement is matching: the number of columns in the first matrix should equal the number of rows in the second matrix.
Consider the matrices in our example:

• Matrix A has dimensions 3 x 4, indicating it has 3 rows and 4 columns.
• Matrix B is a 4 x 3 matrix, with 4 rows and 3 columns.

Since the inner dimensions align (4 columns in A and 4 rows in B), we confirmed that multiplication is possible.
Moreover, the dimensions of the resulting matrix are determined by the outer dimensions: 3 (rows from A) by 3 (columns from B), resulting in a 3x3 matrix.
Grasping this rule is essential to successfully engaging in matrix calculations and achieving correct outcomes.