Problem 18 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The product matrix \(AB\) is \(\begin{bmatrix} -43 & -6 & 56 \\ -7 & 18 & 36 \end{bmatrix}\).

Step 1: Structure of Resultant Matrix

Determine the dimensions of the resulting matrix from the multiplication of matrices \(A\) (2x2) and \(B\) (2x3). The resultant matrix will have dimensions based on the number of rows of the first matrix and the number of columns of the second matrix. Thus, the result will be a 2x3 matrix.

Step 2: Calculate First Row Entries

To find the first row of the resultant matrix, we multiply the first row of \(A\) by each column of \(B\). \(\text{Entry } (1,1): (-1) \times 3 + 5 \times (-8) = -3 - 40 = -43 \\text{Entry } (1,2): (-1) \times 6 + 5 \times 0 = -6 + 0 = -6 \\text{Entry } (1,3): (-1) \times 4 + 5 \times 12 = -4 + 60 = 56 \)

Step 3: Calculate Second Row Entries

To find the second row of the resultant matrix, we multiply the second row of \(A\) by each column of \(B\). \(\text{Entry } (2,1): 3 \times 3 + 2 \times (-8) = 9 - 16 = -7 \\text{Entry } (2,2): 3 \times 6 + 2 \times 0 = 18 + 0 = 18 \\text{Entry } (2,3): 3 \times 4 + 2 \times 12 = 12 + 24 = 36 \)

Step 4: Assemble Resultant Matrix

Combine the computed entries to form the resultant matrix. Thus, the product of matrices \(A\) and \(B\) is: \[AB = \begin{bmatrix} -43 & -6 & 56 \ -7 & 18 & 36 \end{bmatrix} \]

Key Concepts

Understanding Matrix Dimensions

Matrix dimensions are described as the number of rows and columns a matrix has. This is crucial for operations like matrix multiplication. For instance, matrix \(A\) has dimensions 2x2, meaning it has 2 rows and 2 columns. Matrix \(B\) is a 2x3 matrix, indicating 2 rows and 3 columns. When multiplying matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Otherwise, the product is undefined. In our example, since \(A\) has 2 columns and \(B\) has 2 rows, these matrices can be multiplied together.

The dimensions of the resulting matrix are determined by taking the number of rows from the first matrix and the number of columns from the second matrix. Therefore, when multiplying \(A\) and \(B\), we end up with a 2x3 matrix.

Finding the Matrix Product

The matrix product involves a systematic multiplication of rows by columns. When determining the matrix product for \(A\) and \(B\), you perform individual scalar multiplications and then sum the results. Here's how it's done:

• Take each row from the first matrix, \(A\) in this case.
• Then multiply it element-wise with each column of the second matrix (\(B\)), summing the resulting products.

This process is repeated for each row of the first matrix with every column of the second matrix.

For example, the first row of the product of matrices \(A\) and \(B\) is obtained by multiplying and summing the elements from the first row of \(A\) and each column of \(B\). This is then repeated for the second row of \(A\).

The results are assembled into the final matrix product, resulting in a new 2x3 matrix.

Step-by-Step Guide to Matrix Multiplication

Performing matrix multiplication can seem complex, but breaking it down into steps like one might do with a detailed recipe can clarify the process. Here's a guide to multiplying matrices \(A\) and \(B\):

**Step 1: Structure of Resultant Matrix**
Identify the dimensions of the resulting matrix by looking at the dimensions of \(A\) and \(B\). Since \(A\) is 2x2 and \(B\) is 2x3, the resulting matrix will be 2x3.

**Step 2: Calculate First Row Entries**
Focus on the first row of \(A\) and calculate the entries by multiplying it with each column of \(B\). The calculations proceed:

• For column one: \((-1) \times 3 + 5 \times (-8) = -43\)
• For column two: \((-1) \times 6 + 5 \times 0 = -6\)
• For column three: \((-1) \times 4 + 5 \times 12 = 56\)

**Step 3: Calculate Second Row Entries**
Multiply the second row of \(A\) with each column of \(B\):

• For column one: \(3 \times 3 + 2 \times (-8) = -7\)
• For column two: \(3 \times 6 + 2 \times 0 = 18\)
• For column three: \(3 \times 4 + 2 \times 12 = 36\)

**Step 4: Assemble Resultant Matrix**
Combine all the calculated entries to form the final matrix product: \[AB = \begin{bmatrix} -43 & -6 & 56 \-7 & 18 & 36 \end{bmatrix} \]
By following each step, you can efficiently multiply any compatible matrices.