Problem 12 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of multiplying matrix A by 5 is \( \begin{bmatrix} 20 & 30 \\ 65 & 60 \end{bmatrix} \).

Step 1: Understand Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a constant, in this case, the scalar 5.

Step 2: Apply Scalar Multiplication to Matrix A

Matrix \( A \) is given as \( \begin{bmatrix}4 & 6 \13 & 12 \end{bmatrix} \). To find \( 5A \), multiply every element in matrix \( A \) by 5.

Step 3: Multiply Each Element by 5

Calculate the new elements:- First row: \( 5 \times 4 = 20 \) and \( 5 \times 6 = 30 \).- Second row: \( 5 \times 13 = 65 \) and \( 5 \times 12 = 60 \).

Step 4: Write the Resultant Matrix

The resulting matrix after scalar multiplication is: \[ \begin{bmatrix}20 & 30 \65 & 60 \end{bmatrix} \]

Key Concepts

Matrix Operations

Matrix operations are fundamental processes in mathematics that allow us to perform various calculations with matrices. One significant operation within this realm is scalar multiplication. Scalar multiplication involves multiplying each element of a matrix by a constant, referred to as a scalar, which is a real number. This operation is straightforward but critical as it helps scale the matrix either up or down.

When performing scalar multiplication, each entry in the matrix is multiplied by the same scalar. For example, if we have a matrix \( A \) and a scalar \( k \), the resulting matrix \( kA \) will have each element of \( A \) multiplied by \( k \). This operation changes the values of the matrix but preserves its structure, such as the number of rows and columns.

Matrix operations like scalar multiplication form the building blocks for more complex calculations such as matrix addition, subtraction, and multiplication with other matrices. These operations are essential tools used in various fields, including computer graphics, physics, and statistics, allowing for the manipulation of data in a structured form.

Matrix Algebra

Matrix algebra is a powerful mathematical framework that allows us to handle equations and transformations involving matrices. At its core, matrix algebra consists of a set of rules and operations that govern the behavior of matrices. These rules enable us to perform operations like addition, scalar multiplication, and multiplication of matrices.

Matrix algebra is essential in solving linear equations, especially in systems where multiple variables are involved. Scalar multiplication, a vital component of matrix algebra, simplifies handling large sets of equations by scaling entire matrices uniformly. Understanding matrix algebra is crucial for anyone working with linear transformations and data representations in higher dimensions.

Using matrix algebra, we can represent complex systems and perform operations to find solutions efficiently. In fields such as engineering and computer science, matrix algebra underpins algorithms and models that simulate real-world phenomena, making it a cornerstone of modern computational techniques. Diving into matrix algebra opens the door to advanced problem-solving strategies that are applied in various scientific disciplines.

Matrix Calculations

Matrix calculations encompass a variety of techniques used to manipulate matrices to extract information or solve problems. One basic yet essential calculation is scalar multiplication. When you multiply a matrix by a scalar, each element in the matrix is multiplied by this scalar, effectively scaling the entire matrix. This kind of calculation is foundational and aids in performing more intricate matrix operations.

To execute matrix calculations effectively, one must first understand the dimensions of matrices being used, as well as the operations applicable to them. For instance, the result of scalar multiplication will have the same size as the original matrix, but each element's value will change according to the scalar used.

Matrix calculations are vital in numerous applications, from computer graphics, where they are used to transform shapes and models, to systems engineering, where they help in solving complex systems of equations. With a solid grasp of matrix calculations, one can efficiently tackle a wide range of problems in science and mathematics.