Problem 68 - A Graphical Approach to College Algebra

Question

Find each matrix product if possible. $$\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]$$

Step-by-Step Solution

Answer

The product is a 3x1 matrix: \([ax+by+cz, dx+ey+fz, gx+hy+iz]^T\).

Step 1: Determine Dimensions

First, we need to determine the dimensions of the matrices involved. The first matrix is a 3x3 matrix, and the second matrix is a 3x1 matrix.

Step 2: Check Compatibility for Multiplication

For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, the first matrix has 3 columns, and the second matrix has 3 rows, so multiplication is possible.

Step 3: Set Up Matrix Multiplication

Set up the multiplication by taking the dot product of the rows of the 3x3 matrix with the columns of the 3x1 matrix.

Step 4: Calculate the Product

Compute each element of the resulting matrix by multiplying corresponding elements and summing them.The resulting matrix will be:\[\begin{bmatrix} ax + by + cz \ dx + ey + fz \ gx + hy + iz \end{bmatrix}\]

Step 5: Write the Final Product

The result of the matrix multiplication is a 3x1 matrix:- First element: \( ax + by + cz \)- Second element: \( dx + ey + fz \)- Third element: \( gx + hy + iz \)

Key Concepts

Matrix Dimensions

When working with matrices, it's crucial to understand the concept of matrix dimensions. This refers to the number of rows and columns a matrix has. For example, in our exercise, the first matrix is 3x3, meaning it has 3 rows and 3 columns. Similarly, the second matrix is 3x1, having 3 rows and 1 column.
Understanding matrix dimensions helps you set the stage for matrix operations like addition, subtraction, and multiplication. The dimensions of a matrix are typically denoted as "rows × columns." Keep in mind:

• A matrix with "m" rows and "n" columns is termed as an "m x n" matrix.
• The dimensions help in identifying matrix compatibility for operations like multiplication.

By recognizing the dimensions early, you can avoid confusion in later calculations and ensure accuracy in your matrix operations.

Dot Product

The dot product is essential when discussing matrix multiplication. It involves multiplying corresponding elements from one vector (or row of a matrix) with another vector (or column from another matrix) and summing the results. In our problem, each element in the resulting matrix is calculated using a dot product.
Here's how it works:

• Take a row from the first matrix.
• Multiply each of its elements by the corresponding elements in a column from the second matrix.
• Sum these products to form a single element of the product matrix.

So, for the first row of the 3x3 matrix \([a, b, c]\) and the column of the 3x1 matrix \([x, y, z]\), the dot product is computed as \(ax + by + cz\).
This process is repeated for each row of the first matrix to complete the matrix multiplication.

Matrix Compatibility

Matrix compatibility is about determining whether two matrices can be multiplied. The key rule is simple: the number of columns in the first matrix must equal the number of rows in the second matrix.
In our exercise, the first matrix has three columns, and the second matrix has three rows, making them compatible for multiplication. Without this compatibility, the multiplication is not possible. Consider these points:

• Check the dimensions first —it can save time and prevent errors.
• Remembering this rule allows you to confidently proceed with setting up your matrix multiplication.

Knowing about compatibility helps in understanding why sometimes you cannot multiply certain matrices, steering you towards correct mathematical operations.

Resulting Matrix

After confirming that two matrices are compatible and computing the dot products, we arrive at the resulting matrix. This matrix contains the products computed using the dot products in each position.
In the current problem, the final matrix is of size 3x1. It contains:

• The first element is \(ax + by + cz\), derived from the first row of the 3x3 matrix with the column of the 3x1 matrix.
• The second element is \(dx + ey + fz\), calculated from the second row.
• The third element is \(gx + hy + iz\), coming from the third row.

The resulting matrix will always have the number of rows from the first matrix and the number of columns from the second matrix. In this case, 3 rows and 1 column match the dimensions of the resulting matrix as expected.