Problem 15 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of \(-4C\) is \(\begin{bmatrix} -64 & -12 & -28 & -72 \\ -360 & -20 & -12 & -116 \end{bmatrix}\).

Step 1: Identify the Scalar and the Matrix

The task is to multiply the matrix \(C\) by the scalar \(-4\). The given matrix \(C\) is \(\begin{bmatrix} 16 & 3 & 7 & 18 \ 90 & 5 & 3 & 29 \end{bmatrix}\).

Step 2: Multiply Each Element by the Scalar

Multiply each element of matrix \(C\) by the scalar \(-4\). Calculate as follows:\[-4 \times 16 = -64\]\[-4 \times 3 = -12\]\[-4 \times 7 = -28\]\[-4 \times 18 = -72\]For the second row:\[-4 \times 90 = -360\]\[-4 \times 5 = -20\]\[-4 \times 3 = -12\]\[-4 \times 29 = -116\]

Step 3: Construct the Resultant Matrix

Use the results from Step 2 to construct the resultant matrix:\(-4C = \begin{bmatrix} -64 & -12 & -28 & -72 \ -360 & -20 & -12 & -116 \end{bmatrix}\)

Key Concepts

Matrices

Matrices are a fundamental concept in mathematics, especially in fields like linear algebra. They are rectangular arrays of numbers arranged in rows and columns. Each number in a matrix is called an element. The size of a matrix is defined by the number of its rows and columns.For instance, in the matrix \[C=\begin{bmatrix}16 & 3 & 7 & 18 \ 90 & 5 & 3 & 29\end{bmatrix}\]there are two rows and four columns, making it a 2x4 matrix. Matrices can be used to represent a variety of data and perform numerous operations. These operations can involve other matrices or sometimes even single numbers, known as scalars.
Matrix representations help simplify complex systems and allow for easier manipulation using mathematical operations.

Matrix Operations

Matrix operations are essential tools for manipulating matrices. They include actions like addition, subtraction, and multiplication. Specifically, scalar multiplication involves multiplying every element of a matrix by a single number, known as the scalar. This is a type of matrix operation.
Consider the exercise where we multiply matrix \(C\) by the scalar \(-4\). For scalar multiplication, simply multiply each element of the matrix by the scalar. If an element of \(C\) is 16, in scalar multiplication with \(-4\), the result will be \(-64\).Some key points to keep in mind during matrix operations:

• Each element of the matrix is handled independently.
• Scalar multiplication changes the size of each element but not the matrix structure.
• The resultant elements can also be positive or negative, depending on the scalar used.

Resultant Matrix

When you perform operations like scalar multiplication on a matrix, the outcome is called the resultant matrix. This new matrix maintains the same dimensions as the original but has updated elements after the operation. In the given exercise, we multiply matrix \(C\) by \(-4\). After computing, each element of the original matrix is transformed, such as 16 becoming \(-64\).The calculated matrix (resultant matrix) is:\[-4C = \begin{bmatrix} -64 & -12 & -28 & -72 \ -360 & -20 & -12 & -116 \end{bmatrix}\]
Notice that this new matrix still has two rows and four columns. Resultant matrices are handy as they are usually used in subsequent mathematical procedures or real-world applications like physics simulations or economic models. Properly understanding matrices and matrix operations is crucial for developing and solving such complex systems.