Problem 15 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a constant number (the scalar). In this exercise, the scalar is \(-4\) and the matrix in question is \(C\).

Step 2: Identify Matrix C

Matrix \(C\) is given as: \[C = \begin{bmatrix} 16 & 3 & 7 & 18 \ 90 & 5 & 3 & 29 \end{bmatrix}\] where it has 2 rows and 4 columns.

Step 3: Multiply Each Element by -4

To perform \(-4C\), multiply every element in matrix \(C\) by \(-4\):\[\begin{bmatrix}-4 \times 16 & -4 \times 3 & -4 \times 7 & -4 \times 18 \-4 \times 90 & -4 \times 5 & -4 \times 3 & -4 \times 29\end{bmatrix}\]

Step 4: Calculate the Product for Each Element

Perform the calculations for each element:- First row: - \(-4 \times 16 = -64\) - \(-4 \times 3 = -12\) - \(-4 \times 7 = -28\) - \(-4 \times 18 = -72\)- Second row: - \(-4 \times 90 = -360\) - \(-4 \times 5 = -20\) - \(-4 \times 3 = -12\) - \(-4 \times 29 = -116\)

Step 5: Write the Result Matrix

After performing the scalar multiplication, the resulting matrix \(-4C\) is:\[\begin{bmatrix}-64 & -12 & -28 & -72 \-360 & -20 & -12 & -116\end{bmatrix}\]

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in mathematics and specifically in linear algebra. It involves the multiplication of two matrices to produce a third matrix. This operation is not as straightforward as multiplying numbers, as it involves summing the products of elements from the rows of the first matrix and columns of the second matrix.
To perform matrix multiplication:

• The number of columns in the first matrix must match the number of rows in the second matrix.
• Each element in the resulting matrix is calculated by multiplying corresponding elements and summing them up.
• The result is a matrix with a number of rows equal to the rows of the first matrix and columns equal to the columns of the second matrix.

An example: if you multiply a 2x3 matrix by a 3x2 matrix, the result will be a 2x2 matrix. It's important to carefully align rows and columns to ensure the operations are correct. This key operation is used widely in physics, computer graphics, and other fields where systems of equations are modeled.

Matrix Operations

Matrix operations involve different procedures that you can perform on matrices—adding, subtracting, multiplying by a scalar, and more. These operations allow us to manipulate and analyze matrices, which represent systems of equations, vectors, or other data structures.

• Addition and Subtraction: Only matrices of the same dimensions can be added or subtracted by adding or subtracting each corresponding element.
• Scalar Multiplication: Each element in the matrix is multiplied by a scalar value. This operation adjusts the magnitude of the matrix's elements but retains the matrix's overall structure.
• Transposition: The rows of a matrix become its columns and vice versa, useful for various mathematical properties and operations.

Exercise involving operations like scalar multiplication are foundational in transitioning to more complex applications, such as finding inverses or determinants.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between spaces. It is rich in operations and applications, making it essential in various scientific fields. Linear algebra helps solve systems of linear equations, perform transformations, and much more using matrices and vectors. Some key points in linear algebra include:

• Vectors and Spaces: Vectors are more than just lines of numbers; they represent points or directions in space.
• Transformations: Matrices can represent transformations, including rotations, translations, and scaling in various dimensions.
• Determinants and Eigenvalues: These properties of matrices help in solving complex systems and understanding stability in systems dynamics.

The lesson's importance in scalar multiplication serves as a building block in linear algebra, emphasizing the significance of element-by-element operations and set the stage for deeper exploration into solving complex problems and modeling real-world systems.