Problem 22 - College Algebra

Question

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

Step-by-Step Solution

Answer

The product of matrices \( D \) and \( C \) is \( \begin{bmatrix} 74 & 110 \\ 35 & 117 \\ -66 & -42 \end{bmatrix} \).

Step 1: Check Dimensions

Before performing matrix multiplication, confirm that the matrices have dimensions that allow for multiplication. Matrix \( D \) is of size \( 3 \times 3 \) and Matrix \( C \) is of size \( 3 \times 2 \). Since the number of columns in \( D \) equals the number of rows in \( C \), we can multiply them.

Step 2: Set Up the Resultant Matrix

Matrix multiplication of matrices \( D \) \((3 \times 3)\) and \( C \) \((3 \times 2)\) results in a new matrix of size \( 3 \times 2 \). The resultant matrix should have 3 rows and 2 columns.

Step 3: Perform Element-Wise Multiplication for Each Element in the Resultant Matrix

To find each element \( (i, j) \) in the resultant matrix, take the dot product of the \( i^{th} \) row of \( D \) with the \( j^{th} \) column of \( C \).- **Row 1, Column 1**: \( (2 \cdot 4) + (-3 \cdot -2) + (12 \cdot 5) = 8 + 6 + 60 = 74 \)- **Row 1, Column 2**: \( (2 \cdot 10) + (-3 \cdot 6) + (12 \cdot 9) = 20 - 18 + 108 = 110 \)- **Row 2, Column 1**: \( (9 \cdot 4) + (3 \cdot -2) + (1 \cdot 5) = 36 - 6 + 5 = 35 \)- **Row 2, Column 2**: \( (9 \cdot 10) + (3 \cdot 6) + (1 \cdot 9) = 90 + 18 + 9 = 117 \)- **Row 3, Column 1**: \( (0 \cdot 4) + (8 \cdot -2) + (-10 \cdot 5) = 0 - 16 - 50 = -66 \)- **Row 3, Column 2**: \( (0 \cdot 10) + (8 \cdot 6) + (-10 \cdot 9) = 0 + 48 - 90 = -42 \)

Step 4: Write Down the Resulting Matrix

After calculating each element, the resultant matrix from \( D \times C \) is:\[DC = \begin{bmatrix} 74 & 110 \ 35 & 117 \ -66 & -42 \end{bmatrix}\]

Key Concepts

Matrices

Matrices are a collection of numbers arranged in rows and columns, similar to a table or a grid. Each number in a matrix is called an element. The size of a matrix is determined by the number of its rows and columns.
Matrices are often represented inside square brackets or parentheses. For instance, if you look at Matrix A:

• Matrix A: \[\begin{bmatrix}-1 & 5 \3 & 2 \end{bmatrix}\]

This matrix has two rows and two columns, so it is a 2x2 matrix. Matrices come in various sizes depending on the numbers of rows and columns. Understanding the structure of matrices is crucial for performing operations like addition, subtraction, and most importantly, multiplication.

Dot Product

The dot product is a mathematical operation that is central to matrix multiplication. When multiplying matrices, the dot product is used to calculate the elements of the resultant matrix.
Here's how it works: For each element in the resultant matrix, you take one row from the first matrix and one column from the second matrix. You then multiply corresponding elements and sum the results.
Consider calculating an element of the resultant matrix from matrices D and C:

• For example, to find the (1,1) element, multiply the elements in the first row of D by the elements in the first column of C.

This process involves multiplying the first element of the row with the first element of the column, the second element of the row with the second element of the column, and so on, then adding them together to get the dot product. This computed sum gives you the specific element in the resultant matrix.

Matrix Dimensions

Matrix dimensions are expressed as 'rows x columns', referring to the number of rows and columns in the matrix. This concept is particularly important when multiplying matrices, as certain rules govern when matrices can be multiplied.
To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. For example:

• Matrix D has dimensions 3x3, and Matrix C has dimensions 3x2.

Thus, they can be multiplied because the number of columns in Matrix D (which is 3) matches the number of rows in Matrix C (also 3). The resulting matrix from this multiplication will have dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix. In this case, it will be a 3x2 matrix.

Resultant Matrix

The resultant matrix is the product of two matrices being multiplied. Its dimensions, as we previously discussed, depend on the dimensions of the matrices involved in the multiplication.
Once the matrix multiplication is complete, the resultant matrix will contain elements found through the series of dot products, calculated by taking each row in the first matrix and each column in the second matrix.
For matrices D and C from our example, the resultant matrix \(DC\) is:

• \[\begin{bmatrix}74 & 110 \35 & 117 \-66 & -42 \end{bmatrix}\]

Understanding how this matrix is formed and its dimensions is key to mastering matrix multiplication. This resultant matrix is what ultimately represents the combined information from multiplying matrices D and C, reflecting each paired interaction through the dot product method.