Problem 17 - College Algebra

Question

In Exercises \(17-26,\) let $$A=\left[\begin{array}{rr}-3 & -7 \\\2 & -9 \\\5 & 0\end{array}\right] \text { and } B=\left[\begin{array}{rr}-5 & -1 \\\0 & 0 \\\3 & -4\end{array}\right]$$ Solve each matrix equation for \(X\). $$X-A=B$$

Step-by-Step Solution

Answer

The solution of the matrix equation is the matrix \(X = \begin{bmatrix} -8 & -8 \ 2 & -9 \ 8 & -4 \end{bmatrix}\)

Step 1: Understand the problem

We have a matrix equation \(X - A = B\), two given matrices \(A\) & \(B\) of the same order and the task is to solve for matrix \(X\).

Step 2: Matrix Subtraction

In the case of matrix subtraction \(X - A = B\), solving for \(X\) basically entails isolating \(X\) on one side of the equation. This can be done by adding \(A\) to both sides of the equation to cancel out \(-A\) on the left side.

Step 3: Solve for X

After adding matrix \(A\) to both sides of the equation, we get \(X = A + B\). So, to find \(X\), sum matrices \(A\) and \(B\) element-wise. You simply add each element in \(A\) with the corresponding element in \(B\). The resulting matrix will be our solution matrix \(X\).

Step 4: Compute the Result

Matrix \(A = \begin{bmatrix} -3 & -7 \ 2 & -9 \ 5 & 0 \end{bmatrix}\) and matrix \(B = \begin{bmatrix} -5 & -1 \ 0 & 0 \ 3 & -4 \end{bmatrix}\). Adding them gives, \(X = A + B = \begin{bmatrix} -8 & -8 \ 2 & -9 \ 8 & -4 \end{bmatrix}\)

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental concept in linear algebra. It is similar to the way we subtract numbers, but applied to matrices, which are arrays of numbers.
The order of the matrices must be the same to perform matrix subtraction, meaning they both need to have the same number of rows and columns. To subtract one matrix from another, such as in the equation \( X - A = B \), you simply subtract each corresponding element in matrix \( A \) from matrix \( X \).
Each element in matrix \( A \) must align exactly with an element in matrix \( X \) for the subtraction to be valid. Remember, subtraction is just the opposite of addition. When solving \( X - A = B \), it is often useful to add \( A \) to both sides to isolate \( X \).

Matrix Addition

Matrix addition is another core concept in the study of matrices. It involves combining two matrices by adding their corresponding elements.
Just like with matrix subtraction, the matrices must be of the same dimension. When solving the equation \( X = A + B \), each element of the resultant matrix \( X \) is:

• The sum of the corresponding elements from matrix \( A \) and matrix \( B \).

For example, if the elements of turning the first row of matrix \( A \) are -3 and -7, and for matrix \( B \) they're -5 and -1, you add them element-wise to get the first row of matrix \( X \):

• -3 + (-5) = -8
• -7 + (-1) = -8

This element-wise addition must be performed for each and every entry of the matrices. Thus, the resulting matrix combines elements from both initial matrices in a straightforward way.

Solving for X

In linear algebra, solving for \( X \) in equations involving matrices requires strategic manipulation. Consider the equation \( X - A = B \).
The goal is to have \( X \) by itself on one side of the equation. To achieve this, think of matrix subtraction as you would numerical subtraction. To move \( A \) across the equation, you need to do the inverse operation. This means adding \( A \) to both sides of the equation. The equation then becomes \( X = A + B \). This technique helps in isolating \( X \) and turns the problem into a simple addition problem. Once \( X \) is isolated, it becomes clear how to compute the values in \( X \) by executing matrix addition.

Element-wise Addition

Element-wise addition is a key aspect when working with matrices. Unlike regular addition, where single numbers are combined, matrix addition requires adding each corresponding element in two matrices.
Each entry in the resultant matrix is obtained by summing elements that occupy the same position in their respective matrices. Take matrices \( A \) and \( B \) for instance. If matrix \( A \) has elements arranged like this:

• First row: -3 and -7
• Second row: 2 and -9
• Third row: 5 and 0

And matrix \( B \) with elements:

• First row: -5 and -1
• Second row: 0 and 0
• Third row: 3 and -4

The corresponding elements are added together to form matrix \( X \), giving details such as:

• For the first row: -3 + (-5) = -8 and -7 + (-1) = -8
• For the second row: 2 + 0 = 2 and -9 + 0 = -9
• For the third row: 5 + 3 = 8 and 0 + (-4) = -4

This process continues similarly for each subsequent pair of elements in the matrices. This repetitive nature makes it critical to understand each step fully.