Problem 22 - College Algebra

Question

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

Step-by-Step Solution

Answer

The product of matrices \(D\) and \(C\) is \( \left[\begin{array}{rr}74 & 110 \\ 35 & 117 \\ -66 & -42\end{array}\right] \).

Step 1: Verify the Dimensions

Matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. Here, matrix \(D\) is a \(3 \times 3\) matrix and matrix \(C\) is a \(3 \times 2\) matrix. The multiplication \(D \cdot C\) is valid since the number of columns in \(D\) (3 columns) matches the number of rows in \(C\) (3 rows). The result will be a \(3 \times 2\) matrix.

Step 2: Set Up Matrix Multiplication

Matrix multiplication involves taking the dot product of rows from the first matrix with columns from the second matrix. For matrix \(D\) and matrix \(C\), we will calculate each element in the resulting matrix by performing these dot products.

Step 3: Calculate Each Element of the Result Matrix

To find each element \(r_{ij}\) in the resulting matrix, compute the dot product of the \(i^{th}\) row of \(D\) with the \(j^{th}\) column of \(C\): \[\begin{aligned}r_{11} &= (2)(4) + (-3)(-2) + (12)(5) = 8 + 6 + 60 = 74, \r_{12} &= (2)(10) + (-3)(6) + (12)(9) = 20 - 18 + 108 = 110, \r_{21} &= (9)(4) + (3)(-2) + (1)(5) = 36 - 6 + 5 = 35, \r_{22} &= (9)(10) + (3)(6) + (1)(9) = 90 + 18 + 9 = 117, \r_{31} &= (0)(4) + (8)(-2) + (-10)(5) = 0 - 16 - 50 = -66, \r_{32} &= (0)(10) + (8)(6) + (-10)(9) = 0 + 48 - 90 = -42.\end{aligned}\]

Step 4: Construct the Resulting Matrix

After calculating each element, write the resulting matrix by organizing all the calculated values as follows:\[DC = \left[\begin{array}{rr}74 & 110 \ 35 & 117 \ -66 & -42\end{array}\right].\]

Key Concepts

Dot Product

In matrix multiplication, one fundamental operation is the dot product. When multiplying two matrices, each element of the resulting matrix is calculated using the dot product of rows and columns from the original matrices.
For instance, to calculate an element in the first row and first column of the resulting matrix, you must:

• Select the entire first row of the first matrix.
• Select the entire first column of the second matrix.
• Multiply corresponding elements from the selected row and column.
• Sum up all the products to find your desired element.

To give a specific example, say we want to compute the element \( r_{11} \) in the product matrix of matrices \( D \) and \( C \). The operation involves multiplying each element of the first row of \( D \) by each respective element of the first column of \( C \), and then summing up these products. This operation is repeated for each combination of rows and columns, ensuring every element in the resulting matrix is accounted for.

Matrix Dimensions

Before multiplying matrices, understanding matrix dimensions is critical to determine if multiplication is feasible. For any two matrices, say matrix \( A \) and matrix \( B \), matrix multiplication can only occur when the number of columns in \( A \) is equal to the number of rows in \( B \). Otherwise, the operation is not possible.
In this context, matrix \( D \) is a \( 3 \times 3 \) matrix, which means it has 3 rows and 3 columns. On the other hand, matrix \( C \) is a \( 3 \times 2 \) matrix with 3 rows and 2 columns. As the number of columns in \( D \) equals the number of rows in \( C \), multiplication is possible, resulting in a matrix that will have dimensions derived from the first matrix's rows and the second matrix's columns, which is \( 3 \times 2 \) for this example. This results in a product matrix with 3 rows and 2 columns.

Resulting Matrix

After performing matrix multiplication, the resulting matrix is where all calculated values are organized. For matrices \( D \) and \( C \), the resulting matrix reflects the products of the dot products computed in the earlier steps.
This matrix will have dimensions which have already been derived from the number of rows in \( D \) and the number of columns in \( C \). Thus, in this particular multiplication, the result is a \( 3 \times 2 \) matrix.
Following through with the calculations, each entry of the resulting matrix is the sum of products derived from the particular row of one matrix and a column of another. Each element such as \( r_{11}, r_{12}, ... \) is found using this process, organizing them into a structured matrix form:\[DC = \begin{bmatrix}74 & 110 \35 & 117 \-66 & -42\end{bmatrix}\]This concludes the multiplication process, providing the final output of all the calculated element values in a tidy matrix format.