Problem 44 - College Algebra

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: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{ll}{1} & {0} \\ {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{-2} & {3} & {4} \\ {-1} & {1} & {-5}\end{array}\right], C=\left[\begin{array}{rr}{0.5} & {0.1} \\ {1} & {0.2} \\\ {-0.5} & {0.3}\end{array}\right], D=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {-6} & {7} & {5} \\ {4} & {2} & {1}\end{array}\right] $$ $$ D C $$

Step-by-Step Solution

Answer

The product \(DC\) is \(\begin{bmatrix} 1 & -0.2 \\ 1.5 & 2.3 \\ 3.5 & 1.1 \end{bmatrix}\).

Step 1: Check Matrix Dimensions

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(D\) has dimensions \(3 \times 3\) and Matrix \(C\) has dimensions \(3 \times 2\). The number of columns of \(D\) (3) matches the number of rows of \(C\) (3), so the multiplication \(DC\) is possible.

Step 2: Multiply the Matrices

To find the product \(DC\), compute each element of the resulting matrix by taking the dot product of the corresponding row of \(D\) with the corresponding column of \(C\).\[ \text{Element } (1,1): 1 \cdot 0.5 + 0 \cdot 1 + (-1) \cdot (-0.5) = 0.5 + 0.5 = 1 \]\[ \text{Element } (1,2): 1 \cdot 0.1 + 0 \cdot 0.2 + (-1) \cdot 0.3 = 0.1 - 0.3 = -0.2 \]\[ \text{Element } (2,1): (-6) \cdot 0.5 + 7 \cdot 1 + 5 \cdot (-0.5) = -3.0 + 7.0 - 2.5 = 1.5 \]\[ \text{Element } (2,2): (-6) \cdot 0.1 + 7 \cdot 0.2 + 5 \cdot 0.3 = -0.6 + 1.4 + 1.5 = 2.3 \]\[ \text{Element } (3,1): 4 \cdot 0.5 + 2 \cdot 1 + 1 \cdot (-0.5) = 2.0 + 2.0 - 0.5 = 3.5 \]\[ \text{Element } (3,2): 4 \cdot 0.1 + 2 \cdot 0.2 + 1 \cdot 0.3 = 0.4 + 0.4 + 0.3 = 1.1 \]

Step 3: Write the Resulting Matrix

The resulting matrix from the multiplication \(DC\) is:\[\begin{bmatrix}1 & -0.2 \1.5 & 2.3 \3.5 & 1.1\end{bmatrix}\]

Key Concepts

Matrix Dimensions

Matrix dimensions are fundamental in determining whether matrices can be multiplied. Each matrix is described by its number of rows and columns. For example, a matrix with 3 rows and 2 columns is labeled as a 3x2 matrix. This labeling is crucial for understanding if matrix multiplication is possible, as the process requires specific alignment criteria. This rule dictates that the number of columns in the first matrix must match the number of rows in the second matrix. This allows the matrices to "match up" properly so you can calculate the result matrix.

Here's a practical perspective:

• Matrix \(D\) is a 3x3 matrix (3 rows, 3 columns).
• Matrix \(C\) is a 3x2 matrix (3 rows, 2 columns).
• The number of columns in \(D\) matches the number of rows in \(C\), so the multiplication \(DC\) can proceed.

Understanding matrix dimensions will guide you toward recognizing the configuration required for multiplication, ensuring efficient calculations.

Dot Product

The dot product of two vectors is a key concept when multiplying matrices. In matrix multiplication, each element of the resulting matrix is found by computing the dot product of a row vector from the first matrix with a column vector from the second matrix.

The dot product involves:

• Multiplying corresponding elements from the row and column vectors.
• Summing these products to get a single number, which becomes an element in the resulting matrix.

For example, consider calculating element (1,1) in the resulting matrix when multiplying matrices \(D\) and \(C\):
\[\text{Element } (1,1): 1 \cdot 0.5 + 0 \cdot 1 + (-1) \cdot (-0.5) = 0.5 + 0 + 0.5 = 1.0\] The dot product mechanism ensures that matrix operation results accurately reflect the vectors' orientations and alignments, giving you a reliable and methodical approach to matrix multiplication.

Matrix Operations

Matrix operations, especially multiplication, form the basis for various complex calculations in mathematics and applied fields. Understanding the nuances of these operations enables manipulation of datasets, solution of systems of equations, and more.

When performing matrix multiplication, follow these guidelines:

• Ensure you have compatible dimensions for the operation, as explained in the matrix dimensions section.
• Calculate each element of the resulting matrix using the dot product method between rows of the first matrix and columns of the second.
• Double-check your work by verifying each arithmetic step to ensure accuracy.

Steps to write the resulting matrix:

• Compute every element by following the multiplication rules and arranging them according to their row and column positions.
• The resulting matrix for \(DC\) is a 3x2 matrix, meaning 3 rows and 2 columns, as derived from following the steps of the original exercise.

Mastering these operations will allow you to simplify and solve complex mathematical problems effectively, expanding your ability to engage with advanced mathematics.