Problem 21 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The product of matrices \(B\) and \(D\) is \(\left[ \begin{array}{ccc} 60 & 41 & 2 \\ -16 & 120 & -216 \end{array} \right].\)

Step 1: Check Matrix Multiplication Compatibility

To multiply matrices \(B\) and \(D\), we need to ensure that the number of columns in \(B\) is equal to the number of rows in \(D\). Matrix \(B\) is a \(2 \times 3\) matrix, and \(D\) is a \(3 \times 3\) matrix. Since they match (3 columns in \(B\) and 3 rows in \(D\)), the multiplication is possible.

Step 2: Set Up the Resulting Matrix

The resulting matrix from multiplying a \(2 \times 3\) matrix by a \(3 \times 3\) matrix will be a \(2 \times 3\) matrix. Let's denote this resulting matrix as \(R\). It will have the form \(R = \left[ \begin{array}{ccc} r_{11} & r_{12} & r_{13} \ r_{21} & r_{22} & r_{23} \end{array} \right]\).

Step 3: Calculate Element \(r_{11}\)

To find the element \(r_{11}\), multiply the corresponding elements from the first row of \(B\) and the first column of \(D\), and then sum them up: \[ r_{11} = (3 \times 2) + (6 \times 9) + (4 \times 0) = 6 + 54 + 0 = 60. \]

Step 4: Calculate Element \(r_{12}\)

For element \(r_{12}\), multiply the first row of \(B\) by the second column of \(D\) and add: \[ r_{12} = (3 \times -3) + (6 \times 3) + (4 \times 8) = -9 + 18 + 32 = 41. \]

Step 5: Calculate Element \(r_{13}\)

For element \(r_{13}\), multiply the first row of \(B\) by the third column of \(D\) and add: \[ r_{13} = (3 \times 12) + (6 \times 1) + (4 \times -10) = 36 + 6 - 40 = 2. \]

Step 6: Calculate Element \(r_{21}\)

To find the element \(r_{21}\), multiply the second row of \(B\) by the first column of \(D\): \[ r_{21} = (-8 \times 2) + (0 \times 9) + (12 \times 0) = -16 + 0 + 0 = -16. \]

Step 7: Calculate Element \(r_{22}\)

For element \(r_{22}\), multiply the second row of \(B\) by the second column of \(D\): \[ r_{22} = (-8 \times -3) + (0 \times 3) + (12 \times 8) = 24 + 0 + 96 = 120. \]

Step 8: Calculate Element \(r_{23}\)

For element \(r_{23}\), multiply the second row of \(B\) by the third column of \(D\): \[ r_{23} = (-8 \times 12) + (0 \times 1) + (12 \times -10) = -96 + 0 - 120 = -216. \]

Step 9: Combine Results Into Resultant Matrix

Combine all the calculated elements to form the matrix product \(BD\): \[ R = \left[ \begin{array}{ccc} 60 & 41 & 2 \ -16 & 120 & -216 \end{array} \right]. \]

Key Concepts

Matrix Compatibility

Before performing matrix multiplication, we must ensure that the matrices involved are compatible. This is called **Matrix Compatibility**. Two matrices are compatible for multiplication when the number of columns in the first matrix equals the number of rows in the second matrix.

For example, when trying to multiply matrices \(B\) and \(D\) from our exercise, it's important to note:

• Matrix \(B\) is a \(2 \times 3\) matrix, which means it has 2 rows and 3 columns.
• Matrix \(D\) is a \(3 \times 3\) matrix, which means it has 3 rows and 3 columns.

Since the number of columns in \(B\) (3) matches the number of rows in \(D\) (3), these matrices can be multiplied.

Checking matrix compatibility is crucial, as attempting to multiply incompatible matrices will result in an error.

Resultant Matrix

After confirming matrix compatibility, the next step is to determine the dimensions and structure of the **Resultant Matrix**. When multiplying a \(2 \times 3\) matrix by a \(3 \times 3\) matrix, the resultant matrix will be a \(2 \times 3\) matrix.

To understand why, consider:

• The number of rows in the resultant matrix corresponds to the number of rows in the first matrix (which is 2 in this exercise).
• The number of columns in the resultant matrix corresponds to the number of columns in the second matrix (which is 3 in this exercise).

Thus, in our example, the product matrix \(R\) has the form: \[ R = \left[ \begin{array}{ccc} r_{11} & r_{12} & r_{13} \ r_{21} & r_{22} & r_{23} \end{array} \right] \]Each element of the matrix \(R\) is calculated by taking the respective row from the first matrix and column from the second matrix, then summing the products of the elements.

Step by Step Solution

A detailed, **step by step solution** is essential for understanding matrix multiplication, allowing students to follow each calculation closely. Here's how you can break it down:

1. Identify the position for which you want to calculate the value (e.g., \(r_{11}\)).2. Take the corresponding row from the first matrix and column from the second matrix.3. Multiply each pair of corresponding elements.4. Sum all those products to get the value for that position.
For example, for \(r_{11}\), using the first row from \(B\) and the first column from \(D\), the calculation is:\[ r_{11} = (3 \times 2) + (6 \times 9) + (4 \times 0) = 6 + 54 + 0 = 60 \]

Repeat this process for each element of the resulting matrix according to its position. This helps to ensure accuracy and understanding of the mechanics behind matrix multiplication.

Elementary Row Operations

In the realm of matrices, understanding **Elementary Row Operations** is critical. However, it's important to note that they are not directly used in basic matrix multiplication as described in this exercise.

Elementary Row Operations refer to operations you can perform on rows of matrices to simplify or transform them, often utilized in processes like finding the inverse of a matrix or row-reducing matrices to solve linear equations.

These operations include:

• Swapping two rows: Interchanging any two rows.
• Multiplying a row by a non-zero scalar: Scaling all elements in a row by a constant.
• Adding or subtracting a multiple of one row to another row: This helps to eliminate variables and make computation easier.

Although these concepts aren't needed for multiplication like \(B \cdot D\) in our task, they remain fundamental for other matrix techniques and understanding matrix algebra overall.