Problem 22 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Resultant matrix from multiplying \(D\) and \(C\) is \(\left[\begin{array}{rr}74 & 110 \\ 35 & 117 \\ -66 & -42\end{array}\right]\).

Step 1: Setup the Matrix Multiplication

To multiply matrices \(D\) and \(C\), verify that the number of columns in \(D\) equals the number of rows in \(C\). \(D\) is a \(3 \times 3\) matrix, and \(C\) is a \(3 \times 2\) matrix. Since both have "3" in these dimensions, they can be multiplied resulting in a \(3 \times 2\) matrix.

Step 2: Calculate the First Row of the Result Matrix

Calculate the elements of the first row of the resulting matrix by taking the dot product of the first row of \(D\) with each column of \(C\). For the first element: \(2 \cdot 4 + (-3) \cdot (-2) + 12 \cdot 5 = 8 + 6 + 60 = 74\). For the second element: \(2 \cdot 10 + (-3) \cdot 6 + 12 \cdot 9 = 20 - 18 + 108 = 110\).

Step 3: Calculate the Second Row of the Result

Use the second row of \(D\) for calculations. First element: \(9 \cdot 4 + 3 \cdot (-2) + 1 \cdot 5 = 36 - 6 + 5 = 35\). Second element: \(9 \cdot 10 + 3 \cdot 6 + 1 \cdot 9 = 90 + 18 + 9 = 117\).

Step 4: Calculate the Third Row of the Result

Now take the third row of \(D\). First element: \(0 \cdot 4 + 8 \cdot (-2) + (-10) \cdot 5 = 0 - 16 - 50 = -66\). Second element: \(0 \cdot 10 + 8 \cdot 6 + (-10) \cdot 9 = 0 + 48 - 90 = -42\).

Step 5: Form the Resultant Matrix

Combine the results from Steps 2, 3, and 4 to form the matrix resulting from the multiplication of \(D\) and \(C\). Thus, the answer is: \(\left[\begin{array}{rr}74 & 110 \ 35 & 117 \ -66 & -42\end{array}\right]\).

Key Concepts

Matrices

In mathematics, matrices are an array of numbers arranged in rows and columns. They are used to represent data or solve systems of linear equations. Each number in a matrix is referred to as an element or an entry. The size or dimension of a matrix is determined by the number of its rows and columns. For instance, a 2x3 matrix has 2 rows and 3 columns. Matrices are a key concept in linear algebra, and they can be added, subtracted, or multiplied, provided certain conditions are met.
Matrices can represent a wide array of transformations. Here are a few examples:

• Transforming geometric shapes in space
• Modeling graphs and networks
• Solving problems in quantum mechanics

Because matrices can represent complex systems, they are vital tools in fields ranging from physics to computer science.

Dot Product

The dot product is a central part of matrix multiplication. It involves multiplying the corresponding elements of a row from one matrix by elements of a column from another matrix and then adding those products. This operation forms the basis for determining each element in the resultant matrix. Understanding the dot product requires grasping these simple steps:

• Select a row from the first matrix.
• Select a column from the second matrix.
• Multiply the corresponding elements from the row and column.
• Add up the results of these multiplications to get an element of the resultant matrix.

The dot product is why the columns of the first matrix must equal the rows of the second to perform multiplication. This ensures that there is an element match for each dot product operation.

Matrix Dimensions

Matrix dimensions are fundamental to understanding matrix operations, especially multiplication. The dimension of a matrix is defined by the number of rows and columns it has, represented as rows x columns. This terminology is crucial because it determines compatibility for multiplication.
For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix. If matrix A is of dimensions 3x4, it can only be multiplied with a matrix B which is 4xn, where "n" denotes any number of columns in matrix B. The resulting matrix will have the dimensions of the first matrix's rows combined with the second matrix's columns (3xn). Understanding matrix dimensions facilitates identifying compatibility in operations, speeding up complex calculations in practical applications.

Resultant Matrix

The resultant matrix is what you obtain after performing matrix multiplication. It is formed by computing the dot products for all possible pairs of rows from the first matrix and columns from the second matrix. After obtaining a complete set of new elements, they are arranged into a matrix whose dimensions are determined by the number of rows in the first matrix and the number of columns in the second matrix.
Let’s break down the creation of a resultant matrix with matrix multiplication:

• Verify the multiplication conditions: number of columns in the first matrix equals the number of rows in the second matrix.
• Compute each element using the dot product method.
• Arrange computed elements in a grid fashion according to the resultant matrix's dimensions.

The resultant matrix allows one to find solutions for more complicated systems and equations in mathematics and applied sciences.