Problem 24 - College Algebra with Modeling and Visualization

Question

Evaluate the matrix expression. $$\left[\begin{array}{rrr}5 & -1 & 6 \\\\-2 & 10 & 12 \\\5 & 2 & 9\end{array}\right]-\left[\begin{array}{rrr}-1 & 2 & 2 \\\2 & -1 & 2 \\\2 & 2 & -1\end{array}\right]$$

Step-by-Step Solution

Answer

The resultant matrix is \(\begin{bmatrix} 6 & -3 & 4 \\ -4 & 11 & 10 \\ 3 & 0 & 10 \end{bmatrix}\).

Step 1: Write Down the Matrices

Identify and write down the given matrices for the subtraction operation: \[ A = \begin{bmatrix} 5 & -1 & 6 \ -2 & 10 & 12 \ 5 & 2 & 9 \end{bmatrix} \] \[ B = \begin{bmatrix} -1 & 2 & 2 \ 2 & -1 & 2 \ 2 & 2 & -1 \end{bmatrix} \]

Step 2: Perform Element-Wise Subtraction

Subtract the matrices by subtracting corresponding elements from matrix \( B \) from matrix \( A \). Compute as follows for each position (i,j): - \( C_{11} = 5 - (-1) = 5 + 1 = 6 \)- \( C_{12} = -1 - 2 = -3 \)- \( C_{13} = 6 - 2 = 4 \)- \( C_{21} = -2 - 2 = -4 \)- \( C_{22} = 10 - (-1) = 10 + 1 = 11 \)- \( C_{23} = 12 - 2 = 10 \)- \( C_{31} = 5 - 2 = 3 \)- \( C_{32} = 2 - 2 = 0 \)- \( C_{33} = 9 - (-1) = 9 + 1 = 10 \)

Step 3: Write the Resultant Matrix

Combine all the computed elements to form the resultant matrix:\[ C = \begin{bmatrix} 6 & -3 & 4 \ -4 & 11 & 10 \ 3 & 0 & 10 \end{bmatrix} \].

Key Concepts

Matrix Operations

Matrix operations are fundamental procedures in linear algebra and are widely used in various fields such as engineering, physics, and computer science. One of the basic operations on matrices is subtraction, which involves taking two matrices of the same dimensions and subtracting corresponding elements from each matrix.
The given exercise demonstrates this by performing element-wise subtraction on two matrices. In this operation, the matrix elements are subtracted one by one, where the element from the second matrix is subtracted from the element in the first matrix at the same position. This results in a new matrix with the computational product.

• Each entry in the resultant matrix is the difference of the corresponding elements from the two matrices.
• The matrices must be of identical dimensions for subtraction to be possible.

By understanding these basic operations, one can solve more complex matrix equations and engage in more advanced applications in linear algebra.

Elementary Matrices

Elementary matrices play a crucial role in understanding and performing efficient row operations on matrices during various algebraic manipulations. They are used to represent basic transformations that can simplify more complex problems.
For instance, matrix subtraction can be thought of as a series of elementary operations. Although the exercise provided does not explicitly require the use of elementary matrices, recognizing their importance can help in understanding matrix transformations:

• Elementary matrices can be used to swap rows, add multiples of rows, or scale rows.
• These operations are useful in matrix algebra, especially when dealing with systems of linear equations.

A deeper grasp of elementary matrices can enhance one's ability to work with matrix operations more skillfully in various mathematical tasks.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on the study of vectors, vector spaces (also known as linear spaces), linear transformations, and matrices. Understanding linear algebra concepts such as matrix operations and subtractions is crucial for fields like quantum mechanics, statistics, and machine learning.
Within linear algebra, matrix subtraction is just one of the many operations that allow us to solve systems of equations and analyze vectors within linear spaces. This operation is particularly useful in:

• Solving linear systems by reducing matrices to simpler forms.
• Applications in computer graphics where transformations are represented by matrices.
• Economics and other sciences, where data is organized in matrix forms.

Linear algebra provides the foundation for deeper exploration into even more complex mathematical theories and practical applications, making it an essential area of study.