Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(\left.A^{2}=A \cdot A\right)\) \(A=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{rrr}-2 & 3 & 4 \\ -1 & 1 & -5\end{array}\right], C=\left[\begin{array}{rr}0.5 & 0.1 \\ 1 & 0.2 \\ -0.5 & 0.3\end{array}\right], D=\left[\begin{array}{rrr}1 & 0 & -1 \\ -6 & 7 & 5 \\\ 4 & 2 & 1\end{array}\right]\) \(D C\)

Step-by-Step Solution

Answer

The product matrix \(DC\) is \(\left[ \begin{array}{cc} 1 & -0.2 \\ -1.5 & 1.9 \\ 3.5 & 1.3 \end{array} \right]\).

Step 1: Determine the dimensions

Matrix \(D\) is a \(3 \times 3\) matrix and matrix \(C\) is a \(3 \times 2\) matrix. For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix, which is true here since \(D\) has 3 columns and \(C\) has 3 rows.

Step 2: Set up the matrix multiplication

To multiply matrix \(D\) by matrix \(C\), calculate each element of the product by taking the dot product of the rows of \(D\) with the columns of \(C\). The resulting matrix will be a \(3 \times 2\) matrix.

Step 3: Calculate the elements of the product matrix

Calculate each element of the resulting matrix:1. First row, first column: \([1 \cdot 0.5 + 0 \cdot 1 + (-1) \cdot (-0.5)] = 1\)2. First row, second column: \([1 \cdot 0.1 + 0 \cdot 0.2 + (-1) \cdot 0.3] = -0.2\)3. Second row, first column: \([-6 \cdot 0.5 + 7 \cdot 1 + 5 \cdot (-0.5)] = -1.5\)4. Second row, second column: \([-6 \cdot 0.1 + 7 \cdot 0.2 + 5 \cdot 0.3] = 1.9\)5. Third row, first column: \([4 \cdot 0.5 + 2 \cdot 1 + 1 \cdot (-0.5)] = 3.5\)6. Third row, second column: \([4 \cdot 0.1 + 2 \cdot 0.2 + 1 \cdot 0.3] = 1.3\)

Key Concepts

matrix dimensions

When working with matrices, understanding their dimensions is crucial for performing operations like multiplication. A matrix is simply an array of numbers organized in rows and columns. The dimensions of a matrix are given in the form \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.
For example, matrix \((D)\) from the exercise has dimensions \(3 \times 3\) because it consists of 3 rows and 3 columns. Similarly, matrix \((C)\) has dimensions \(3 \times 2\), indicating 3 rows and 2 columns. Understanding the structure of matrices is the first step in performing any matrix operation, as the dimensions dictate how matrices interact with each other.
In matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. This requirement ensures that the rows of the first matrix align correctly with the columns of the second matrix, making multiplication possible.

dot product

The dot product is a foundational concept in matrix multiplication. It involves multiplying corresponding elements and then summing those products. This operation is applied to a row of the first matrix and a column of the second matrix to produce an individual element in the resulting matrix.
For example, when multiplying two matrices, each element of the product matrix is calculated using the dot product. If you have a row from the first matrix \(( [a, b] )\) and a column from the second matrix \(( [c, d] ) )\), the resulting value would be \( (a \cdot c + b \cdot d)\).
In the given problem, to compute each element of matrix \(DC\), the dot product is taken between rows of \(D\) and columns of \(C\). This operation is repeated for each position in the resulting \(3 \times 2\) matrix, filling in values one by one.

matrix operations

Matrix operations involve several activities like addition, subtraction, and more complex procedures like transposition and multiplication. Each operation has specific rules based on the dimensions and arrangement of matrices.
Focus lies especially on multiplication in this exercise. It is not simply a matter of multiplying corresponding elements. Instead, it requires carefully pairing rows with columns and using the dot product. Other operations like addition or subtraction require the matrices to be of the same dimension.
So, if you're asked to multiply matrices \(D\) and \(C\), you first ensure their dimensions allow for this. Then, systematically calculate each element of the resulting matrix. Understanding how different operations work prepares you to solve various matrix-related problems smoothly. Each matrix operation adheres to distinct mathematical rules, which helps in solving equations and real-world problems.

product matrix calculation

The calculation of a product matrix from multiplication is a step-by-step procedure. Begin with ensuring matrices' dimensions allow multiplication: the first matrix's columns must match the second matrix's rows.
For instance, matrix \(D\) is \(3 \times 3\) and matrix \(C\) is \(3 \times 2\). These matrices can be multiplied because \(D\) has 3 columns and \(C\) has 3 rows. The resulting matrix will adopt the dimensions \(3 \times 2\).
To compute the elements of this resulting matrix:

• Multiply each row element of matrix \(D\) by the corresponding column element of matrix \(C\).
• For each product, sum up these multiplications for every entity in the product matrix.
• Repeat this across all required row-column pairs until the new matrix is completely filled.

In the exercise example, each result exemplifies this process, such as calculating \(1 \cdot 0.5 + 0 \cdot 1 + (-1) \cdot (-0.5)\) to get an individual element of matrix \(DC\). Ensure accurate calculations to form a precise product matrix.