Problem 43 - Precalculus Mathematics for Calculus

Question

Let $$\begin{array}{l} A=\left[\begin{array}{cccc} 1 & 0 & 6 & -1 \\ 2 & \frac{1}{2} & 4 & 0 \end{array}\right] \quad C=\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ -2 \end{array}\right] \\ B=\left[\begin{array}{llll} 1 & 7 & -9 & 2 \end{array}\right] \end{array}$$ Determine which of the following products are defined, and calculate the ones that are. $$\begin{aligned} &A B C \quad A C B \quad B A C\\\ &B C A \quad C A B \quad C B A \end{aligned}$$

Step-by-Step Solution

Answer

Only product \( B C \) is defined, resulting in 6.

Step 1: Determine if A B C is defined

To determine if the product \( A B C \) is defined, we must check the dimensions of each matrix. \( A \) is a \( 2 \times 4 \) matrix, \( B \) is a \( 1 \times 4 \) matrix, and \( C \) is a \( 4 \times 1 \) matrix. For matrix multiplication to be possible, the inner dimensions of the matrices must match. Thus for \( A B \), the multiplication is only possible if the number of columns in \( A \) equals the number of rows in \( B \). Since they do not (4 vs. 1), \( A B C \) is not defined.

Step 2: Determine if A C B is defined

The product \( A C B \) involves multiplying \( A (2\times4) \), \( C (4\times1) \), and \( B (1\times4) \). To find \( A C \), ensure the number of columns in \( A \) matches the number of rows in \( C \), which they do, allowing a \( 2 \times 1 \) result. Then ensure the number of columns in the result (1) matches the rows in \( B \), which they don't (1 vs. 4), rendering \( A C B \) undefined.

Step 3: Determine if B A C is defined

We check \( B A C \) by starting with \( B (1\times4) \) and \( A (2\times4) \). The multiplication \( B A \) isn't possible because the number of columns in \( B \) doesn't match the number of rows in \( A \) (4 vs. 2). Therefore, \( B A C \) is undefined.

Step 4: Determine if B C A is defined and calculate

First, find \( B C \), where \( B (1\times4) \) times \( C (4\times1) \) is possible if the inner dimensions match, which they do, producing a \( 1 \times 1 \) result.\[B C = [1\; 7\; -9\; 2] \times \begin{bmatrix} 1 \ 0 \ -1 \ -2 \end{bmatrix} = 1(1) + 7(0) - 9(-1) + 2(-2) = 1 + 0 + 9 - 4 = 6\]Next, multiply the \( 1 \times 1 \) result with \( A (2 \times 4) \), yielding another mismatch in dimensions, thus \( B C A \) is undefined.

Step 5: Determine if C A B is defined

First, check \( C A \) where \( C (4\times1) \) and \( A (2\times4) \) are involved. The number of columns in \( C \) doesn't match the number of rows in \( A \) (1 vs. 4), thus \( C A \) is not possible. Therefore, \( C A B \) is undefined.

Step 6: Determine if C B A is defined

Consider \( C (4\times1) \), \( B (1\times4) \), \( A (2\times4) \). Multiplications \( C B \) and \( B A \) both aren't possible because of unmatched inner dimensions (1 vs. 4, 4 vs. 2), so \( C B A \) is undefined.

Key Concepts

Matrix Dimensions

When discussing matrix multiplication, understanding matrix dimensions is crucial. A matrix is a rectangular array of numbers, and its size is described using dimensions. The dimensions are written as "rows x columns," indicating the number of rows and columns present in a matrix. For example, matrix \( A \) from our problem has dimensions of \( 2 \times 4 \). This means matrix \( A \) has 2 rows and 4 columns.
Understanding these dimensions is essential in matrix multiplication. The key rule to multiply two matrices successfully is that the number of columns in the first matrix must be the same as the number of rows in the second matrix. For instance, when multiplying a \( 2 \times 4 \) matrix by a \( 4 \times 1 \) matrix, the multiplication is feasible because the "4s" (the columns of the first and the rows of the second) match.
Matrix dimensions are the gatekeepers to determine if two matrices can be multiplied. If the inside dimensions don't match, the operation isn't defined. This principle guides all decisions regarding multiplying matrices in this exercise.

Matrix Product

The matrix product, also known as matrix multiplication, involves taking two matrices and producing a third matrix as the result. The product is valid only when the matrices involved have compatible dimensions. Specifically, if you have matrices \( A \) with dimensions \( m \times n \) and \( B \) with dimensions \( n \times p \), their product \( AB \) results in a new matrix of dimensions \( m \times p \).
In our exercise, checking the feasibility of each product is the first step. For instance, when computing \( BC \), \( B \) has dimensions \( 1 \times 4 \) and \( C \) has \( 4 \times 1 \). The number of columns in \( B \) equals the number of rows in \( C \), making the multiplication possible.
The resulting matrix from multiplying \( B \) and \( C \) is a \( 1 \times 1 \) matrix, which is a single number here: 6. Each element in the resulting product matrix is calculated by multiplying corresponding elements and summing these products. Understanding matrix product entails ensuring the existing matrices can "communicate" through aligned dimensions and sums of products.

Matrix Operations

Matrix operations encompass more than just multiplication. However, in this exercise, our focus is on multiplication due to matrix dimensions. When two matrices are multiplied (if dimensions match), each row of the first matrix is multiplied by each column of the second.For example, when evaluating expression like \( AB \), the process involves:

• Taking each row element in the first matrix and multiplying it by corresponding column elements in the second.
• Summing the products to form an element in the resulting matrix.

Such operations require precision as they follow strict mathematical rules regarding order and dimensions. Moreover, these operations are not always commutative, meaning \( AB eq BA \) generally.
If dimensions aren't compatible (e.g., trying to combine \( A \) with \( B \) in our exercise), the operation isn't possible. Thus, a strong grasp of how these operations function is crucial in determining when a matrix product can be computed, providing a broader view of matrix operations that can be conducted based on given matrices.