Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$C A$$

Step-by-Step Solution

Answer

The multiplication \(C \cdot A\) is not possible due to dimension mismatch.

Step 1: Determine Matrix Dimensions

Identify the dimensions of each matrix to ensure matrix multiplication is possible. Matrix \(C\) has dimensions \(2 \times 3\), and matrix \(A\) has dimensions \(2 \times 2\). Since the number of columns in \(C\) is not equal to the number of rows in \(A\), the multiplication \(C \cdot A\) is not defined.

Step 2: Conclude Multiplication Not Possible

As the matrix dimensions are not compatible for multiplication, \(C \cdot A\) cannot be computed. Multiplication of matrices is only possible if the number of columns in the first matrix equals the number of rows in the second matrix.

Key Concepts

Matrix Dimensions

When dealing with matrices, understanding their dimensions is essential. Matrix dimensions are expressed as "rows x columns." This means that if a matrix has dimensions of \(2 \times 3\), it has 2 rows and 3 columns. For example, in our exercise, matrix \(C\) has dimensions \(2 \times 3\), while matrix \(A\) has dimensions \(2 \times 2\). It's crucial to clearly identify the dimensions because they play a fundamental role in determining what operations are possible between matrices. Without knowing the dimensions, you cannot proceed with operations like matrix multiplication.

Matrix Compatibility

Matrix compatibility is primarily concerned with whether two matrices can be multiplied together. This depends on their dimensions. Specifically, to multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. For instance, if you have two matrices with dimensions of \(2 \times 3\) and \(3 \times 2\), they can be multiplied because the middle dimensions (3 in the first matrix and 3 in the second) are the same. In the exercise, we found that matrices \(C\) and \(A\) are not compatible because matrix \(C\)'s columns do not match matrix \(A\)'s rows. As a result, their multiplication is undefined, demonstrating how important it is to ensure compatibility before attempting matrix operations.

Matrix Operations

Matrix operations include addition, subtraction, and multiplication. Each has its own rules and conditions. Among these, multiplication is often the most complex. Not only must the matrices be dimensionally compatible, but the process of multiplying them involves taking dot products of rows and columns.

Here's a brief run-through of matrix multiplication:

• Ensure the number of columns in the first matrix equals the number of rows in the second matrix.
• Multiply elements from rows of the first matrix with corresponding elements from columns of the second matrix.
• Sum these products to fill in the resulting matrix.

This operation results in a new matrix with dimensions that combine the outer dimensions of the matrices involved. While the process might seem daunting initially, practice and a firm grasp of core matrix concepts make matrix operations more intuitive.