Problem 46 - Precalculus: Mathematics for Calculus

Question

Solve for \(x\) and \(y\). $$\left[\begin{array}{rr}x & y \\ -y & x\end{array}\right]-\left[\begin{array}{rr}y & x \\ x & -y\end{array}\right]=\left[\begin{array}{rr}4 & -4 \\ -6 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

\(x = 5\), \(y = 1\)

Step 1: Set Up the Equation

To solve the matrix equation, we first write out both sides and set them equal:\[\begin{bmatrix} x & y \ -y & x \end{bmatrix} - \begin{bmatrix} y & x \ x & -y \end{bmatrix} = \begin{bmatrix} 4 & -4 \ -6 & 6 \end{bmatrix}\]The left side can be simplified by subtracting each corresponding element in the matrices.

Step 2: Simplify the Matrices

Subtract the elements of the second matrix from the first:\[\begin{bmatrix} x-y & y-x \ -y-x & x-(-y) \end{bmatrix} = \begin{bmatrix} 4 & -4 \ -6 & 6 \end{bmatrix}\]This results in:\[\begin{bmatrix} x-y & y-x \ -y-x & x+y \end{bmatrix}\]

Step 3: Set Up and Solve the Element Equations

Now, match each element of the resulting matrix to the corresponding element in the equality matrix:1. \(x-y = 4\)2. \(y-x = -4\)3. \(-y-x = -6\)4. \(x+y = 6\)Solve these equations individually.

Step 4: Solve First Equation

Solve the equation \(x - y = 4\):\[x = y + 4\]

Step 5: Solve Second Equation

Use the second equation \(y - x = -4\):Substitute \(x = y + 4\):\[y - (y + 4) = -4\]\[-4 = -4\]This confirms the relationship found in the first equation.

Step 6: Solve Third Equation

Use the third equation \(-y - x = -6\).From Step 4, replace \(x\):\[-y - (y + 4) = -6\]Simplify and solve:\[-2y - 4 = -6\]\[-2y = -2\]\[y = 1\]

Step 7: Solve Fourth Equation

Now substitute \(y = 1\) into \(x + y = 6\):\[x + 1 = 6\]\[x = 5\]

Step 8: Verify the Solution

Substitute \(x = 5\) and \(y = 1\) back into the original equation:Check each element to ensure they give the provided output matrix:\[\begin{bmatrix} 5 & 1 \ -1 & 5 \end{bmatrix} - \begin{bmatrix} 1 & 5 \ 5 & -1 \end{bmatrix} = \begin{bmatrix} 4 & -4 \ -6 & 6 \end{bmatrix}\]The verification shows that these values satisfy the original equation.

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental operation in the area of matrix algebra. It involves subtracting one matrix from another. This operation is done element-wise, which means we will subtract each corresponding element in the matrices involved.
This process is similar to regular subtraction in arithmetic. However, it is important to highlight that both matrices must be of the same dimensions for the subtraction to be valid.
For example, if we have two matrices, \( A \) and \( B \), both being 2x2 matrices, matrix subtraction is performed by calculating \( A - B \) as follows:

• Subtract the element in the first row and first column of \( B \) from the same element in \( A \).
• Continue this for each corresponding element in \( A \) and \( B \), placing the results in a new matrix of the same dimensions.

In our original exercise, matrix subtraction simplifies to a new matrix, facilitating solving the system of equations.

Systems of Linear Equations

A system of linear equations consists of equations with multiple variables. These variables are solved simultaneously to find their values. By solving these systems, we can identify the values of unknowns that satisfy all the equations in the system.
In mathematical notation, a simple system might look like this:

• Equation 1: \( ax + by = c \)
• Equation 2: \( dx + ey = f \)

In our matrix-centered exercise, once we gather all the simplified equations from the subtraction, we essentially have a system of linear equations. Each element comparison results in a simple linear equation, like \( x - y = 4 \) and \( x + y = 6 \). To solve such a system:

• Express one variable in terms of another using one of the equations.
• Substitute this expression into the other equation to solve for one variable.
• With the found value, substitute back to compute the second variable.

This strategy, known as substitution, is a conventional approach to solving linear systems.

Verification of Solutions

Verification is crucial to ensure the solution provided satisfies the initial problem. In algebra and matrix equations, it confirms that our calculated values for the variables are correct and fit within the original model. For this step, we substitute the solution back into the original equations or matrices.
In a systematic verification process, once the solutions for \( x \) and \( y \) are derived and appear plausible, replace these in the matrices. For our exercise:

• First, compute the left-hand side, using the inferred values: \( x = 5 \) and \( y = 1 \).
• Verify if subtracting the right matrix from the left matrix produces the matrix given on the right-hand side.

Thus, a match between the computed and observed matrices confirms the solutions' correctness. Verification reassures us that our mathematical steps and logical reasoning were executed properly.