Problem 18 - 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 \(X\). $$x-B=A$$

Step-by-Step Solution

Answer

To solve the matrix equation \(X - B = A\) for \(X\), add matrix \(B\) to both sides of the equation to get \(X = A + B\). The matrix \(X\) is then found by adding the corresponding elements of matrices \(A\) and \(B\) together.

Step 1: Analyze the Given Equation

The given equation is \(X - B = A\). We are tasked with finding matrix \(X\). Since matrix addition and subtraction conform to the basic rules of arithmetic, we can manipulate the equation to isolate \(X\) in a similar manner to how we would do so with real numbers, by adding \(B\) to both sides.

Step 2: Add Matrix B to Both Sides

Perform the operation \(X - B + B = A + B\), yielding \(X = A + B\). This is because \(B - B\) cancels out on the left side of the equation, leaving us with \(X\).

Step 3: Perform Matrix Addition on the Right Side

Now, evaluate \(A + B\) by adding corresponding elements from matrices \(A\) and \(B\). That is, add each element in the same position in \(A\) to the corresponding element in the same position in \(B\). This is the standard rule for matrix addition. It will yield a new matrix, which is our solution matrix \(X\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation where you combine two matrices by adding their corresponding elements together. To perform matrix addition, both matrices involved must have the same dimensions, which means they should have the same number of rows and columns.
The process is quite straightforward:

• Identify the corresponding elements in each matrix. These are the elements that exist at the same position in the matrix layout.
• Add these elements together to form a new matrix, which will also have the same dimensions.

In our example, matrices \(A\) and \(B\) both have the dimensions 3x2 (3 rows and 2 columns). This means we can add them directly. For instance, for the element in the first row and first column, you would add \(-3\) from matrix \(A\) to \(-5\) from matrix \(B\), giving \(-8\). This process is repeated for each element in the matrices.
Matrix addition is used in various applications such as in solving matrix equations, finding statistics in datasets, and even in computer graphics.

Matrix Subtraction

Just like matrix addition, matrix subtraction involves taking two matrices and subtracting their corresponding elements. It requires both matrices to have identical dimensions.

• Identify the elements of each matrix that share the same position.
• Subtract each element of the second matrix from the corresponding element of the first matrix.

In our specific exercise, the initial equation given is \(X - B = A\). Instead of directly calculating a subtraction, we aim to manipulate the equation so \(X\) is isolated. The matrix subtraction here forms part of our strategy to isolate \(X\): by adding \(B\) to both sides, we effectively "cancel" \(B\) on the left-hand side, thanks to the rule that any matrix subtracted from itself results in a zero matrix. This logical manipulation underpins solving matrix equations where subtraction is present.

Isolation of Variable

The main goal of solving matrix equations, or any equation, is often to isolate the variable. In this context, the variable is the matrix \(X\). Isolation refers to having the variable on one side of the equation, free from any other matrices.
In our exercise, we are working with the equation \(X - B = A\). To isolate \(X\), we use the inverse operation of subtraction, which is addition. By adding matrix \(B\) to both sides of the equation, we do the following:

• On the left-hand side, \(B\) cancels itself out (\(B - B = 0\)), effectively leaving us with \(X\) on one side of the equation.
• The right-hand side becomes \(A + B\), which we then calculate using matrix addition.

This operation highlights a strategy often employed in algebra—using inverse operations to simplify and solve for variables. Isolating the variable is a powerful technique across different types of mathematical problems, providing clarity and direction in finding solutions.