Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Find \(A B\) $$ A=\left[\begin{array}{rrrr} 2 & 1 & 0 & -3 \\ -7 & 0 & -2 & 4 \end{array}\right], \quad B=\left[\begin{array}{rrr} 4 & -2 & 0 \\ 1 & 1 & -2 \\ 0 & 0 & 5 \\ -3 & -1 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The product matrix \( AB \) is \( \begin{bmatrix} 19 & -2 & -2 \\ -40 & 10 & -10 \end{bmatrix} \).

Step 1: Verify Matrix Dimensions

First, verify if the matrices can be multiplied. Matrix \( A \) is a \(2 \times 4\) matrix, and matrix \( B \) is a \(4 \times 3\) matrix. Since the number of columns in \( A \) is equal to the number of rows in \( B \), they can be multiplied.

Step 2: Set Up the Resultant Matrix

The resultant matrix \( AB \) from multiplying a \(2 \times 4\) matrix with a \(4 \times 3\) matrix is a \(2 \times 3\) matrix. Therefore, \( AB \) will have 2 rows and 3 columns.

Step 3: Calculate Each Element of the Resultant Matrix

Compute each element of the resultant matrix \( AB \) using the formula: \( (AB)_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj} \).- For the element at the first row and first column, compute: \( 2 \times 4 + 1 \times 1 + 0 \times 0 + (-3) \times (-3) = 19 \).- For the element at the first row and second column, compute: \( 2 \times (-2) + 1 \times 1 + 0 \times 0 + (-3) \times (-1) = -2 \).- For the element at the first row and third column, compute: \( 2 \times 0 + 1 \times (-2) + 0 \times 5 + (-3) \times 0 = -2 \).- For the element at the second row and first column, compute: \( -7 \times 4 + 0 \times 1 + (-2) \times 0 + 4 \times (-3) = -40 \).- For the element at the second row and second column, compute: \( -7 \times (-2) + 0 \times 1 + (-2) \times 0 + 4 \times (-1) = 10 \).- For the element at the second row and third column, compute: \( -7 \times 0 + 0 \times (-2) + (-2) \times 5 + 4 \times 0 = -10 \).

Step 4: Construct the Resultant Matrix AB

Using the computed elements, the resultant matrix \( AB \) is:\[AB = \begin{bmatrix} 19 & -2 & -2 \ -40 & 10 & -10 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is essential for performing matrix multiplication. Each matrix is described by its number of rows and columns. For example, Matrix \( A \) has dimensions of \( 2 \times 4 \), meaning it has 2 rows and 4 columns. Matrix \( B \) is a \( 4 \times 3 \) matrix, with 4 rows and 3 columns. The rule for multiplying two matrices is that the number of columns in the first matrix must match the number of rows in the second matrix. In our example, matrix \( A \) can be multiplied by matrix \( B \) because the number of columns in \( A \) (4 columns) matches the number of rows in \( B \) (4 rows).

This property allows us to proceed with the multiplication process to produce a new matrix. This basic compatibility check, often skipped, is crucial as it determines if the multiplication is even possible.

Resultant Matrix

The resultant matrix stems from the multiplication of two compatible matrices. Once the matrices are verified to be multipart-compatible, we can determine the dimensions of the resultant matrix. For matrices \( A \) and \( B \) as described earlier, the resultant matrix \( AB \) will have dimensions \( 2 \times 3 \).

The number of rows in the resultant matrix is equal to the number of rows in the first matrix (\( A \) with 2 rows). Meanwhile, the number of columns in the resultant matrix is equal to the number of columns in the second matrix (\( B \) with 3 columns).

Thus, the resultant matrix from multiplying these two would have 2 rows and 3 columns. This new matrix represents the sum of products of specific elements from each matrix, organized in a structured form that highlights the relationships between the original datasets.

Elements Calculation

Calculating each element of the resultant matrix is possibly the most engaging part of matrix multiplication. Each element at position \( (i,j) \) in the resultant matrix \( AB \) is calculated using the formula: \((AB)_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj} \). This means that for each element, you multiply corresponding elements from the row of the first matrix and the column of the second matrix, and then sum these products.

Let's break it down:

• To find the element at the first row and first column: Compute \( 2 \times 4 + 1 \times 1 + 0 \times 0 + (-3) \times (-3) = 19 \).
• Similarly, compute each subsequent element in this manner, such as \( 2 \times (-2) + 1 \times 1 + 0 \times 0 + (-3) \times (-1) = -2 \) for the first row, second column, and so on.

By using the sum of products method for each element, students can visualize how each part of the matrices contributes to the outcome. This systematic approach ensures all calculations are organized and precise.

Matrix Operations

Matrix operations include a variety of techniques for handling matrices in mathematical contexts, with multiplication being one of the key operations. Before matrix multiplication, always ensure dimensional compatibility, as discussed earlier. Once verified, set up the operations by considering each component's interaction and contribution to the resultant matrix.

Matrices are multiplied row-by-column, meticulously creating each element through the sum of products. This process not only combines numerical values but also highlights patterns and structures within datasets.

• Row-by-column method focuses on how separate matrix portions interlock to create complex, informative structures.
• This operation is at the heart of matrix analysis, frequently used in fields ranging from computer graphics to statistical modeling.

Mastering these operations enables broader applications and insights, allowing students to approach more complex problems with confidence and clarity.