Problem 14 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

-2B = \( \begin{bmatrix} -6 & -18 \\ -42 & -24 \\ 0 & -128 \end{bmatrix} \)

Step 1: Identify the Matrix and Scalar

We need to perform scalar multiplication on matrix \(B\) by \(-2\). The matrix \(B\) is given by: \[ B = \begin{bmatrix} 3 & 9 \ 21 & 12 \ 0 & 64 \end{bmatrix} \] We will multiply each element of \(B\) by the scalar \(-2\).

Step 2: Multiply Each Element

For scalar multiplication, multiply each element of the matrix by the scalar \(-2\). Perform the following calculations for each element: \(-2 \times 3 = -6\)\(-2 \times 9 = -18\)\(-2 \times 21 = -42\)\(-2 \times 12 = -24\)\(-2 \times 0 = 0\)\(-2 \times 64 = -128\)

Step 3: Construct the Result Matrix

Using the calculated values, construct the resulting matrix after scalar multiplication: \[ -2B = \begin{bmatrix} -6 & -18 \ -42 & -24 \ 0 & -128 \end{bmatrix} \]

Key Concepts

Matrix Operations

Matrix operations are fundamental procedures in algebra that allow us to manipulate matrices to solve various mathematical problems. At their core, matrix operations include addition, subtraction, scalar multiplication, multiplication, and sometimes division, though division is not as straightforward as the others due to matrices not being directly divisible.

• Addition and Subtraction: These operations can be performed only on matrices of the same size. They involve adding or subtracting corresponding elements from each matrix.
• Scalar Multiplication: This involves multiplying every element of a matrix by a scalar (a constant number). This operation is the focal point of our exercise, where we multiplied matrix \(B\) by \(-2\).
• Matrix Multiplication: This is more complex and requires the number of columns in the first matrix to equal the number of rows in the second matrix.

The outcome of such operations often helps in solving systems of equations and other algebraic functions effectively.

Matrix Multiplication

Matrix multiplication is a fundamental technique used in linear algebra, enabling us to combine matrices in a way that the structure and relationships across elements are preserved. It's crucial to remember that not any two matrices can be multiplied. For two matrices \(A\) and \(B\), matrix multiplication is defined when the number of columns in \(A\) matches the number of rows in \(B\). Here's how it works:

• Each element of the resulting matrix is obtained by taking the dot product of the rows of the first matrix with the columns of the second matrix.
• The result is a new matrix where the number of rows corresponds to the first matrix, and the number of columns corresponds to the second matrix.
• This operation is usually not commutative, meaning \(A \times B\) does not necessarily equal \(B \times A\).

Thus, matrix multiplication allows for transformations and combinations of datasets that are represented in grid format—extremely useful in fields such as physics, computer science, and statistics.

Matrices in Algebra

Matrices are a powerful tool in algebra used to solve systems of equations, perform transformations, and represent data succinctly and efficiently. They allow complex mathematical operations to be conducted easily, often transforming and simplifying problems that involve multiple variables.
In algebra, matrices can represent various real-world situations and processes, such as:

• Solving simultaneous equations using techniques like Gaussian elimination or Cramer's rule.
• Encoding transformations used in computer graphics, involving scaling, rotating, and translating images.
• Handling data sets in statistics and machine learning, where variables and observations can be organized in matrices to simplify computations.

Utilizing matrices leads to deeper insights into mathematical relationships and provides a versatile method to address numeric and algebraic problems.