Problem 44 - Precalculus: Mathematics for Calculus

Question

Solve for \(x\) and \(y\). $$3\left[\begin{array}{ll}x & y \\ y & x\end{array}\right]=\left[\begin{array}{rr}6 & -9 \\ -9 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

The solution is \(x = 2\) and \(y = -3\).

Step 1: Write the system of equations

We begin by equating each element of the left matrix with its corresponding element in the right matrix, since the matrices must be equal.The matrix equation is: \[3\begin{bmatrix} x & y \ y & x \end{bmatrix} = \begin{bmatrix} 6 & -9 \ -9 & 6 \end{bmatrix}\]From this, we can derive the individual equations:1. \(3x = 6\)2. \(3y = -9\)3. \(3y = -9\) (This is a repetition and confirms consistency)4. \(3x = 6\) (Another repetition that shows symmetry and consistency)We will solve these equations for \(x\) and \(y\).

Step 2: Solve for x

Using the equation \(3x = 6\) from Step 1, solve for \(x\):Divide both sides by 3:\[x = \frac{6}{3}\]Therefore,\[x = 2\]

Step 3: Solve for y

Using the equation \(3y = -9\) from Step 1, solve for \(y\):Divide both sides by 3:\[y = \frac{-9}{3}\]Therefore,\[y = -3\]

Step 4: Verify the solution

Substitute \(x = 2\) and \(y = -3\) back into the original equations to ensure consistency.\[3\begin{bmatrix} 2 & -3 \ -3 & 2 \end{bmatrix} = \begin{bmatrix} 6 & -9 \ -9 & 6 \end{bmatrix}\]Calculate the left matrix by multiplying each element by 3:\[\begin{bmatrix}3 \times 2 & 3 \times (-3) \3 \times (-3) & 3 \times 2\end{bmatrix} = \begin{bmatrix} 6 & -9 \ -9 & 6 \end{bmatrix}\]Both sides match, confirming that \(x = 2\) and \(y = -3\) is correct.

Key Concepts

Solving Systems of Equations

When faced with a system of equations, you are essentially dealing with multiple equations simultaneously. These equations often contain more than one variable, and the goal is to find values for these variables that satisfy all equations in the system. In the context of matrices, solving systems of equations becomes visually intuitive because each entry in the matrix corresponds to an equation derived from the equality of matrices.
For instance, in the exercise, we have a matrix equation that provides us two distinct equations when we equate each element from the resulting matrix:

• For the first equation, the upper-left elements are taken: \(3x = 6\).
• The second equation, derived from the upper-right (and confirmed by lower-left) elements, is \(3y = -9\).

By solving these equations simultaneously, you arrive at solutions for \(x\) and \(y\). Here, solving means finding a value for \(x\) and \(y\) that makes both equations true at the same time. This highlights the power of matrix equations in organizing and simplifying the solving of systems.

Matrix Equality

Matrix equality is a fundamental concept which states that two matrices are equal if and only if each corresponding element in the matrices is identical. This can be visualized by comparing both matrices side-by-side and ensuring that every element at each position matches.
In the exercise, matrix equality provided the system of equations to be solved. The matrix on the left needs to be equal to the matrix on the right for the given equation to hold true. Each element comparison gave rise to a separate equation:

• Comparing top left, \(3x\) must equal 6.
• And from both the top right and bottom left, \(3y\) must equal -9.

The idea here is to utilize matrix equality to reduce a potentially complex problem into linear equations that can be more straightforward to solve. This principle is vital when dealing with any problems in Linear Algebra involving matrices and helps in understanding higher dimensional expressions by their components.

Elementwise Operations

Elementwise operations in matrices involve executing arithmetic operations such as addition, subtraction, or multiplication on corresponding elements across matrices. For any operation to be applicable, two matrices need to be of the same dimension.
In our example, we deal specifically with elementwise multiplication during matrix scaling. The matrix on the left is being scaled by multiplying each of its elements by 3, which must then align elementwise with the matrix on the right for the equation to hold true:

• Top left element: \(3 \times x = 6\)
• Top right and bottom left elements: \(3 \times y = -9\)
• Bottom right element: \(3 \times x = 6\), which is consistent with the top left element.

Understanding elementwise operations is crucial as it allows you to perform operations across entire matrices, simplifying complex calculations. This concept is utilized in many advanced mathematical applications, including computer graphics and data analysis, making it important to grasp fully.