Problem 16 - College Algebra

Question

Use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{rrrr}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) \(\frac{1}{2} C\)

Step-by-Step Solution

Answer

The scaled matrix is \(\left[\begin{array}{rrrr}8 & 1.5 & 3.5 & 9 \\ 45 & 2.5 & 1.5 & 14.5\end{array}\right]\)."

Step 1: Understanding Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar value. In this exercise, we are asked to perform scalar multiplication on matrix \(C\) using the scalar \(\frac{1}{2}\). This means each element of matrix \(C\) will be halved.

Step 2: Apply Scalar to First Row of Matrix C

Matrix \(C\) is given by \(C=\left[\begin{array}{rrrr}16 & 3 & 7 & 18 \ 90 & 5 & 3 & 29\end{array}\right]\). Compute the first row: \(\frac{1}{2} \times 16 = 8\), \(\frac{1}{2} \times 3 = 1.5\), \(\frac{1}{2} \times 7 = 3.5\), \(\frac{1}{2} \times 18 = 9\). The resulting first row is \([8, 1.5, 3.5, 9]\).

Step 3: Apply Scalar to Second Row of Matrix C

Proceed to the second row calculations: \(\frac{1}{2} \times 90 = 45\), \(\frac{1}{2} \times 5 = 2.5\), \(\frac{1}{2} \times 3 = 1.5\), \(\frac{1}{2} \times 29 = 14.5\). The resulting second row is \([45, 2.5, 1.5, 14.5]\).

Step 4: Combine Rows for Scaled Matrix

Combine the rows obtained in Steps 2 and 3 to form the resultant scaled matrix. Thus, \(\frac{1}{2} C = \left[\begin{array}{rrrr}8 & 1.5 & 3.5 & 9 \ 45 & 2.5 & 1.5 & 14.5\end{array}\right]\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra. It involves multiplying each element of a matrix by a single number, known as a scalar. Let's break this down with a simple example.

Imagine you have a matrix filled with numbers, and you want to "scale" it. Scaling means you're either increasing or decreasing each element by a specific factor—this factor is your scalar. For example, using the scalar \( \frac{1}{2} \), you would take each element of a matrix and multiply it by \( \frac{1}{2} \).

This operation is straightforward:

• It's consistent for every element.
• It doesn't change the shape of the matrix.
• Every element is touched by the same scalar value.

An essential tip to remember is that while scalar multiplication can make your entire matrix larger or smaller, it retains the proportionality of elements.

Matrix Operations

Matrix operations are a set of mathematical procedures that involve matrices. Beyond scalar multiplication, there are a few other operations crucial to understanding matrices.

Some vital matrix operations include:

• **Addition:** Where two matrices of the same dimensions combine by adding their corresponding elements.
• **Subtraction:** Similar to addition but involves subtracting corresponding elements.
• **Multiplication:** More complex than scalar multiplication; it involves combining elements by rows and columns to form a new matrix.

Each of these operations follows specific mathematical rules. It's crucial to pay attention to the properties of matrices involved, such as their dimensions. For example, matrix multiplication requires specific dimension compatibility: the number of columns in the first matrix must equal the number of rows in the second.

Understanding these operations gives a solid foundation for more advanced topics in linear algebra.

Element-wise Multiplication

Element-wise multiplication, also known as the Hadamard product, is a matrix operation where each element of one matrix is multiplied by the corresponding element of another matrix. It's different from regular matrix multiplication, which involves a dot product.

This type of multiplication is defined only for matrices of the same size:

• Each element in the same position within their respective matrices is multiplied together.
• The result is a new matrix of the same size.
• It's an operation that maintains the individuality of each element’s product.

Consider two matrices, \(A\) and \(B\), both having dimensions \(2 \times 2\):
\[ A = \begin{pmatrix} a_{1,1} & a_{1,2} \ a_{2,1} & a_{2,2} \end{pmatrix}, B = \begin{pmatrix} b_{1,1} & b_{1,2} \ b_{2,1} & b_{2,2} \end{pmatrix} \]

The element-wise multiplication \(A \odot B\) results in:
\[ \begin{pmatrix} a_{1,1} \cdot b_{1,1} & a_{1,2} \cdot b_{1,2} \ a_{2,1} \cdot b_{2,1} & a_{2,2} \cdot b_{2,2} \end{pmatrix} \]
Mastering this concept can be particularly useful for computations in data processing and image manipulation, where element-wise operations are prevalent.