Problem 19 - College Algebra

Question

For the following exercises, use the matrices below to perform matrix multiplication. $$ A=\left[\begin{array}{rr}{-1} & {5} \\ {3} & {2}\end{array}\right], B=\left[\begin{array}{rrr}{3} & {6} & {4} \\ {-8} & {0} & {12}\end{array}\right], C=\left[\begin{array}{rr}{4} & {10} \\ {-2} & {6} \\\ {5} & {9}\end{array}\right], D=\left[\begin{array}{rrr}{2} & {-3} & {12} \\\ {9} & {3} & {1} \\ {0} & {8} & {-10}\end{array}\right] $$ $$ B C $$

Step-by-Step Solution

Answer

\[ BC = \left[\begin{array}{cc} 20 & 102 \\ 28 & 28 \end{array}\right] \]

Step 1: Understand Matrix Dimensions

The order of matrices determines whether multiplication is possible. Matrix \( B \) is a \( 2 \times 3 \) matrix, and matrix \( C \) is a \( 3 \times 2 \) matrix. Since the number of columns in \( B \) matches the number of rows in \( C \), matrix multiplication is possible, resulting in a \( 2 \times 2 \) matrix.

Step 2: Perform Multiplication for First Row, First Column

Multiply the first row of matrix \( B \) with the first column of matrix \( C \) and sum the products: \[ (3)(4) + (6)(-2) + (4)(5) = 12 - 12 + 20 = 20 \]

Step 3: Perform Multiplication for First Row, Second Column

Multiply the first row of matrix \( B \) with the second column of matrix \( C \) and sum the products: \[ (3)(10) + (6)(6) + (4)(9) = 30 + 36 + 36 = 102 \]

Step 4: Perform Multiplication for Second Row, First Column

Multiply the second row of matrix \( B \) with the first column of matrix \( C \) and sum the products: \[ (-8)(4) + (0)(-2) + (12)(5) = -32 + 0 + 60 = 28 \]

Step 5: Perform Multiplication for Second Row, Second Column

Multiply the second row of matrix \( B \) with the second column of matrix \( C \) and sum the products: \[ (-8)(10) + (0)(6) + (12)(9) = -80 + 0 + 108 = 28 \]

Step 6: Compile Resulting Matrix

Compile the results from Steps 2 through 5 into the resulting matrix:\[ BC = \left[\begin{array}{cc} 20 & 102 \ 28 & 28 \end{array}\right] \]

Key Concepts

Matrices

In mathematics, matrices are a critical tool for organizing and manipulating numerical data. A matrix is essentially a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, in this exercise, each matrix (like matrix A, B, C, and D) has its unique shape and dimensions. Organizing data in this way allows for various mathematical operations, such as matrix addition, multiplication, and more, which are fundamental in fields like physics, engineering, computer science, and economics.
Understand that each element of a matrix can be used in calculations to solve complex systems of equations efficiently. The placement, order, and type of numbers in a matrix define how these operations can be performed.

Matrix Dimensions

Understanding the dimensions of a matrix is crucial because it dictates the matrix's operations compatibility. Dimensions are expressed as 'rows by columns,' for instance, a matrix with 3 rows and 2 columns is a 3 x 2 matrix, which you read as "three by two."
Let's consider matrices B and C in our exercise. Matrix B is a 2 x 3 matrix, and matrix C is a 3 x 2 matrix. For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix. Here, matrix B's 3 columns match matrix C's 3 rows, allowing multiplication. The result of this operation will be a 2 x 2 matrix, providing a new perspective and form from the original matrices.

• The resultant matrix's first dimension comes from the first matrix's row number.
• The second dimension comes from the second matrix's column number.

Step by Step Solution

Matrix multiplication might seem complex at first, but a step-by-step approach simplifies the process. We'll look at how you can multiply matrices B and C from our exercise in clear steps:

1. **Check Compatibility**: Ensure dimensions match for multiplication. For B (2 x 3) and C (3 x 2), multiplication is possible.
2. **Calculate Individual Elements**: Focus on each element of the resulting matrix by working row by column.

• For the element in row 1, column 1: Multiply and sum as follows (3)(4) + (6)(-2) + (4)(5) = 20.
• For row 1, column 2: (3)(10) + (6)(6) + (4)(9) = 102.
• Row 2, column 1: (-8)(4) + (0)(-2) + (12)(5) = 28.
• Row 2, column 2: (-8)(10) + (0)(6) + (12)(9) = 28.

3. **Assemble the Result**: The answers provide the new matrix: \[BC = \left[\begin{array}{cc} 20 & 102 \ 28 & 28 \end{array}\right]\] Each calculation here involves multiplying corresponding elements and finding their sum.

Educational Content

Providing educational content involves breaking down concepts into digestible information. With matrices and operations like matrix multiplication, it's essential to walk learners through the logic and process. We strive to make complex mathematical operations feel manageable and straightforward.
Start by familiarizing students with matrix definitions and operations. Use relatable analogies or practical examples in engineering or computer graphics to show matrices' real-world applications. Then, employ visual aids or digital tools to demonstrate step-by-step approaches, ensuring comprehension and retention by illustrating the mechanics visibly.
Continuous engagement with practice problems helps reinforce these ideas, solidifying the learner's ability to apply these techniques confidently.