Problem 51 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B,\) and \(C\) are defined as follows. $$A=\left[\begin{array}{rrrr}1 & 0 & 6 & -1 \\ 2 & \frac{1}{2} & 4 & 0\end{array}\right]$$ $$B=\left[\begin{array}{llll}1 & 7 & -9 & 2\end{array}\right] \quad C=\left[\begin{array}{r}1 \\ 0 \\ -1 \\ -2\end{array}\right]$$ 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

The products \(A C\), \(B C\), and \(C B\) are defined. Calculations: \(A C = \begin{bmatrix} -3 \\ -2 \end{bmatrix}\), \(B C = 6\), \(C B = \begin{bmatrix} 1 & 7 & -9 & 2 \\ 0 & 0 & 0 & 0 \\ -1 & -7 & 9 & -2 \\ -2 & -14 & 18 & -4 \end{bmatrix}\).

Step 1: Understanding the Problem

We need to determine which matrix products among \(A B C\), \(A C B\), \(B A C\), \(B C A\), \(C A B\), and \(C B A\) are defined and calculate them.

Step 1: Determine the Dimensions of Each Matrix

Matrix \(A\) is \(2 \times 4\), matrix \(B\) is \(1 \times 4\), and matrix \(C\) is \(4 \times 1\). To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.

Step 2: Check If Products Are Defined

1. In \(A B\), the dimensions \(2 \times 4\) and \(1 \times 4\) do not allow multiplication as the inner dimensions don't match.2. In \(A C\), the dimensions \(2 \times 4\) and \(4 \times 1\) allow multiplication resulting in a \(2 \times 1\) matrix.3. In \(B A\), the dimensions \(1 \times 4\) and \(2 \times 4\) do not allow multiplication as the inner dimensions don't match.4. In \(B C\), the dimensions \(1 \times 4\) and \(4 \times 1\) allow multiplication resulting in a \(1 \times 1\) matrix.5. In \(C A\), the dimensions \(4 \times 1\) and \(2 \times 4\) do not allow multiplication as the inner dimensions don't match.6. In \(C B\), the dimensions \(4 \times 1\) and \(1 \times 4\) allow multiplication resulting in a \(4 \times 4\) matrix.

Step 3: Compute Defined Products

1. **Compute \(A C\):** \[ A C = \begin{bmatrix} 1 & 0 & 6 & -1 \ 2 & \frac{1}{2} & 4 & 0 \end{bmatrix} \begin{bmatrix} 1 \ 0 \ -1 \ -2 \end{bmatrix} = \begin{bmatrix} 1(1) + 0(0) + 6(-1) + (-1)(-2) \ 2(1) + \frac{1}{2}(0) + 4(-1) + 0(-2) \end{bmatrix} = \begin{bmatrix} 1 - 6 + 2 \ 2 - 4 \end{bmatrix} = \begin{bmatrix} -3 \ -2 \end{bmatrix} \]2. **Compute \(B C\):** \[ B C = \begin{bmatrix} 1 & 7 & -9 & 2 \end{bmatrix} \begin{bmatrix} 1 \ 0 \ -1 \ -2 \end{bmatrix} = 1(1) + 7(0) + (-9)(-1) + 2(-2) = 1 + 9 - 4 = 6 \]3. **Compute \(C B\):** \[ C B = \begin{bmatrix} 1 \ 0 \ -1 \ -2 \end{bmatrix} \begin{bmatrix} 1 & 7 & -9 & 2 \end{bmatrix} = \begin{bmatrix} 1(1) & 1(7) & 1(-9) & 1(2) \ 0(1) & 0(7) & 0(-9) & 0(2) \ -1(1) & -1(7) & -1(-9) & -1(2) \ -2(1) & -2(7) & -2(-9) & -2(2) \end{bmatrix} = \begin{bmatrix} 1 & 7 & -9 & 2 \ 0 & 0 & 0 & 0 \ -1 & -7 & 9 & -2 \ -2 & -14 & 18 & -4 \end{bmatrix} \]

Step 5: Conclusion: List All Results

Products that are defined are \(A C\), \(B C\), and \(C B\). - \(A B\), \(B A\), and \(C A\) are not defined.- Calculated results: - \(A C = \begin{bmatrix} -3 \ -2 \end{bmatrix}\) - \(B C = 6\) - \(C B = \begin{bmatrix} 1 & 7 & -9 & 2 \ 0 & 0 & 0 & 0 \ -1 & -7 & 9 & -2 \ -2 & -14 & 18 & -4 \end{bmatrix}\)

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial for performing matrix multiplication. Each matrix has dimensions described in terms of rows and columns. For instance, if a matrix has 3 rows and 4 columns, its dimensions are stated as 3 × 4 (read as "3 by 4"). Different operations require that matrices comply with specific dimension rules.

In multiplication, the key rule is that the number of columns in the first matrix must match the number of rows in the second matrix.

• Matrix A is a 2 × 4 matrix, meaning it has 2 rows and 4 columns.
• Matrix B is a 1 × 4 matrix, consisting of 1 row and 4 columns.
• Matrix C is 4 × 1, with 4 rows and 1 column.

Recognizing these dimensions helps in quickly evaluating whether the matrices can be multiplied to produce a valid result.

Matrix Products

Matrix products refer to the results we get when matrices are multiplied according to compatible dimensions. It is important to verify dimensions before attempting to multiply.

Based on our understanding of matrix dimensions:

• For the product AC, matrix A (2 × 4) can be multiplied by matrix C (4 × 1), resulting in a 2 × 1 matrix.
• For the product BC, matrix B (1 × 4) can be multiplied by matrix C (4 × 1), producing a 1 × 1 matrix.
• For the product CB, matrix C (4 × 1) is multiplied by matrix B (1 × 4), and the result is a larger 4 × 4 matrix.

For other combinations such as AB, BA, and CA, the inner dimensions don't match, and thus, these products aren't defined. Thus, understanding matrix products requires careful attention to the dimensions to ensure valid multiplication.

Matrix Operations

Matrix operations include addition, subtraction, and multiplication, each with rules to follow based on matrix dimensions. Among these, multiplication is the most complex as it relies heavily on inner dimensions matching.

Let's highlight a few operations conducted in the exercise:

• **Compute AC:** The resulting 2 × 1 matrix from multiplying A (2 × 4) with C (4 × 1).
• **Compute BC:** Multiplying B (1 × 4) with C (4 × 1) gives a 1 × 1 matrix.
• **Compute CB:** The result is a 4 × 4 matrix after multiplying C (4 × 1) with B (1 × 4).

These operations follow a systematic multiplication process where each element in the resulting matrix is computed from the corresponding row of the first matrix and the column of the second. Exact results will depend on the specific values within each matrix. Matrix operations, particularly multiplication, are fundamental in various mathematical, engineering, and computer science applications. Understanding how to properly manipulate these will enable you to apply matrix algebra effectively.