Problem 57 - A Graphical Approach to College Algebra

Question

Find each matrix product if possible. $$\left[\begin{array}{rrr} 3 & -4 & 1 \\ 5 & 0 & 2 \end{array}\right]\left[\begin{array}{r} -1 \\ 4 \\ 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The product of the matrices is \( \left[ \begin{array}{r} -17 \\ -1 \end{array} \right] \).

Step 1: Verify Compatibility for Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Here, the first matrix is of size 2x3, i.e., 2 rows and 3 columns, and the second matrix is of size 3x1, i.e., 3 rows and 1 column. Therefore, matrix multiplication is possible as the number of columns in the first matrix (3) matches the number of rows in the second matrix (3).

Step 2: Perform the Matrix Multiplication

To find the product, multiply each element of the rows of the first matrix by the corresponding elements of the column of the second matrix and then sum the results.- For the first element of the product matrix:\[ (3)(-1) + (-4)(4) + (1)(2) = -3 - 16 + 2 = -17 \]- For the second element of the product matrix:\[ (5)(-1) + (0)(4) + (2)(2) = -5 + 0 + 4 = -1 \]

Step 3: Write the Resulting Product Matrix

Combine the results found in the previous step to form the product matrix. The resulting matrix from the multiplication is:\[ \left[ \begin{array}{r} -17 \ -1 \end{array} \right] \]

Key Concepts

Matrix Product

A matrix product involves multiplying two matrices together to produce a new matrix. This process is also called matrix multiplication. It is a fundamental operation in linear algebra and is widely used in various fields, such as computer graphics and engineering.

Matrix multiplication is not as straightforward as multiplying single numbers. Here’s why:

• You don't multiply matching elements. Instead, you take each row of the first matrix and multiply it by each column of the second matrix.
• The resulting matrix, called the product matrix, will have a size determined by the number of rows from the first matrix and the number of columns in the second matrix.
• Each element of the product matrix is calculated by summing up the products of the corresponding row elements from the first and column elements from the second matrix.

Understanding how to execute a matrix product correctly is vital for solving problems involving matrices.

Matrix Compatibility

Matrix compatibility is key to determining whether two matrices can be multiplied. Not all matrices are compatible for multiplication, which stems from their dimensions (or size).

To check if two matrices are compatible, follow these rules:

• Look at the number of columns in the first matrix — this must equal the number of rows in the second matrix.
• If this condition is met, you can proceed with the multiplication.
• If not, multiplication is not possible.

In our example exercise, we have two matrices: one that's 2x3 and another that's 3x1. Since the first has 3 columns and the second has 3 rows, they are compatible.

This rule ensures that each element in a row of the first matrix can be "paired" with an element in a column of the second matrix to form a valid mathematical operation.

Matrix Operations

Matrix operations encompass various mathematical processes, including addition, subtraction, and multiplication. In this context, we've focused on a specific operation known as matrix multiplication.

Let's highlight some of the critical aspects of matrix operations:

• Commutativity: Unlike simple arithmetic, matrix multiplication is not commutative. This means that \(A \cdot B\) does not necessarily equal \(B \cdot A\).
• Associativity: Matrix multiplication is associative, meaning that \(A \cdot (B \cdot C) = (A \cdot B) \cdot C\).
• Distributivity: It also holds the distributive property. For example, \(A \cdot (B + C) = A \cdot B + A \cdot C\).

Understanding these operations is crucial because they underpin the rules and procedures for manipulating matrices.

Mastery of these concepts allows one to compute matrix products effectively and use matrices for more complex calculations and real-world applications.