Problem 20 - Algebra 1

Question

Find the difference of the matrices. $$ \left[\begin{array}{rrr} -5 & 11 & -2 \\ -10 & 4 & 6 \end{array}\right]-\left[\begin{array}{rrr} -3 & 0 & 2 \\ 8 & -5 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The difference of the given matrices is \[ \left[\begin{array}{rrr} -2 & 11 & -4 \ -18 & 9 & 7 \end{array}\right] \]

Step 1: Identify the Given Matrices

We have two matrices: Matrix A = \[ \left[\begin{array}{rrr} -5 & 11 & -2 \ -10 & 4 & 6 \end{array}\right] \] Matrix B = \[ \left[\begin{array}{rrr} -3 & 0 & 2 \ 8 & -5 & -1 \end{array}\right] \]

Step 2: Perform Subtraction for Each Element

Subtract the corresponding elements in the two matrices. This means subtracting the first element of Matrix B from the first element of Matrix A, and so on. We do this for all elements. Let us represent the difference matrix by C.

Step 3: Calculate the Resultant Matrix

Now, conduct the operation: C = A - B = \[ \left[\begin{array}{rrr} (-5) - (-3) & 11 - 0 & (-2) - 2 \ (-10) - 8 & 4 - (-5) & 6 - (-1) \end{array}\right] = \[ \left[\begin{array}{rrr} -2 & 11 & -4 \ -18 & 9 & 7 \end{array}\right] \].

Key Concepts

Matrices

A matrix is a collection of numbers arranged into a fixed number of rows and columns. It is a rectangular array in which each item, called an element, is identified by two indexes: the first one for the row and the second one for the column. Matrices are used in various fields, including physics, computer science, statistics, and more. They function as a tool to solve systems of linear equations, transform geometric figures, represent graphs, and even encode and efficiently handle complex data structures.

In the given exercise, two matrices are in focus, each with two rows and three columns, often referred to as a 2x3 matrix. The elements of these matrices can be real numbers, as in this case, or they can consist of variables or more complex expressions. When working with matrices, it's important to remember that their dimensions must be the same when performing element-wise operations like addition or subtraction.

Algebraic Operations

Algebraic operations are the foundation of algebra and include addition, subtraction, multiplication, division, and exponentiation. They help solve equations and understand the relationships between variables. When it comes to matrices, these operations have specific rules. Matrix addition and subtraction are straightforward, they involve adding or subtracting corresponding elements of the matrices. However, multiplication and division are more complex, requiring a set of rules for operations to occur. For instance, you can multiply a matrix by a scalar or by another matrix, but dimensions must align properly (the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication). Division between matrices isn't defined, but we use the concept of a matrix inverse to solve problems similar to division.

In our exercise, we're focusing on matrix subtraction, which strictly requires both matrices to have the same dimensions (the same number of rows and columns), allowing for the direct subtraction of their corresponding elements.

Elementary Operations

Elementary operations are basic computations that are performed on matrices, belonging to the set of allowable operations that can be used for solving systems of equations or for simplifying matrices in linear algebra. They include row and column additions, scalar multiplication of rows or columns, and row or column interchanges. These operations are crucial when reducing a matrix to its row-echelon form or when performing matrix inversion using methods like Gaussian elimination.

In the context of our matrix subtraction exercise, we perform an elementary operation element-wise. We calculate the difference for each pair of corresponding elements in matrices A and B. This involves simple subtraction, which is an example of an elementary operation. As we progress through such elementary operations carefully, ensuring that we match the corresponding elements and operate on them correctly, we obtain a new matrix C, which is the result of the subtraction A - B.