Problem 20 - Precalculus: Mathematical for Calculus

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as $$\begin{array}{l} A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right] \\ D=\left[\begin{array}{rrr} 7 & 3 \end{array}\right] & E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \\ F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \quad G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$5 A$$

Step-by-Step Solution

Answer

The result of \( 5A \) is \( \begin{bmatrix} 10 & -25 \\ 0 & 35 \end{bmatrix} \).

Step 1: Understand Scalar Multiplication

The expression requires multiplying the matrix \( A \) by the scalar 5. Scalar multiplication involves multiplying each element of the matrix by the scalar value.

Step 2: Multiply Each Element by the Scalar

Matrix \( A \) is given as \( \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \). To find \( 5A \), multiply each element by 5: \[ 5 \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 5 \times 2 & 5 \times -5 \ 5 \times 0 & 5 \times 7 \end{bmatrix} = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}. \]

Key Concepts

Scalar Multiplication

Scalar multiplication is a basic operation in linear algebra, particularly when working with matrices. It involves multiplying every element of a matrix by a single number, known as a scalar. This is akin to scaling the entire matrix by that value.
When performing scalar multiplication, it's essential to systematically apply the scalar to each entry in the matrix. Here's how it works step-by-step:

• First, take each element from the original matrix.
• Then, multiply that element by the scalar.
• Replace the original element with the resulting product.

For instance, if we have a matrix \(A\) and a scalar \(5\), each component of \(A\) is multiplied by \(5\). If \(A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\), then \(5A = \begin{bmatrix} 5 \times 2 & 5 \times -5 \ 5 \times 0 & 5 \times 7 \end{bmatrix} = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\).
Remember, scalar multiplication is straightforward: each matrix entry is independently affected by the scalar, yielding a new matrix of the same dimensions and pattern.

Matrices

Matrices are an arrangement of numbers into rows and columns, resembling a grid. Each number within a matrix is called an element. Matrices are fundamental in mathematics and various applications:

• Mathematics: They are used in solving systems of linear equations, transformations, and more.
• Computer Science: They form the basis for graphical transformations, networking algorithms, and machine learning.
• Physics and Engineering: Matrices help describe physical systems and their transformations.

A matrix is generally denoted by a capital letter like \(A\), and its size is given as 'm x n' where \(m\) is the number of rows and \(n\) the number of columns. For example, the matrix \(A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\) is a 2x2 matrix, meaning it has 2 rows and 2 columns.
Matrices can vary greatly in dimensions, with special forms like square matrices (equal number of rows and columns) and identity matrices (which act as the multiplicative identity in matrix algebra). Understanding how to organize and manipulate matrices is crucial for advanced mathematical concepts.

Matrix Algebra

Matrix algebra is a broad area that encompasses various operations on matrices. Under matrix algebra, we perform operations like addition, subtraction, multiplication, and several other computations involving matrices. Each operation has specific rules, which makes matrix algebra a structured and fascinating topic.
With matrix addition and subtraction, matrices of the same dimensions are required. This is because operations occur on corresponding elements. Matrix multiplication, on the other hand, is more complex. It involves a combination of corresponding elements, and the resulting matrix may have different dimensions than the original matrices.
Matrix algebra is not just limited to operations within individual matrices, but also encompasses transformations and the manipulation of multiple matrices to solve complex problems or represent multi-dimensional data effectively. As you delve deeper into matrix operations, you'll uncover many fascinating possibilities for representing and solving problems across various fields. Understanding these fundamentals is key to mastering more advanced concepts in linear algebra.