Problem 36 - 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

\(x = 2\) and \(y = -3\).

Step 1: Equate the Matrices

To solve the problem, equate the corresponding elements from the left matrix and the right matrix, as both matrices must be equal. This gives us two equations: 1. \(3x = 6\)2. \(3y = -9\)

Step 2: Solve for x

Take the first equation \(3x = 6\). Divide both sides by 3 to isolate \(x\): \[x = \frac{6}{3} = 2\]

Step 3: Solve for y

Take the second equation \(3y = -9\). Divide both sides by 3 to isolate \(y\): \[y = \frac{-9}{3} = -3\]

Step 4: Check the Solution

Substitute \(x = 2\) and \(y = -3\) back into the matrices to verify. Both sides should result in the same matrix: Left hand side:\[3\begin{bmatrix} 2 & -3 \ -3 & 2 \end{bmatrix} = \begin{bmatrix} 6 & -9 \ -9 & 6 \end{bmatrix}\]This verifies our solution is correct.

Key Concepts

Matrices

Matrices are an essential mathematical tool for organizing numbers, symbols, or expressions into rows and columns. They are usually represented within square brackets. Each element in a matrix can be a real number, a variable, or even a complex number.
Matrices are used in various fields such as computer graphics, statistics, and engineering to handle data more efficiently.

• Rows are horizontal, and columns are vertical.
• The size of a matrix is commonly referred to as its dimensions (number of rows by number of columns).
• A matrix can represent system equations or data points in a graph.

Understanding matrices can make solving systems of equations simpler and more efficient. Using matrices can help visualize and manipulate the data they contain. Being comfortable with matrices is a powerful skill in both academic and professional settings.

Solving Systems of Equations

When dealing with systems of equations, a matrix can simplify the process of finding solutions. A system of equations is a collection of two or more equations with the same set of unknowns. The task is to find values for these unknowns that satisfy all the equations simultaneously.
Matrices can represent these systems compactly:

• Each row in the matrix can correspond to an equation in the system.
• Each element in a row corresponds to a coefficient in the equation.

In our specific exercise, the matrix on the left represents a set of linear equations. By equating this matrix to another, we have effectively created multiple equations that we can solve one at a time. This process highlights the versatility and practicality of matrices in solving systems of equations.

Equating Matrices

Equating matrices is a straightforward yet powerful concept. When two matrices of the same dimensions are equated, it means each element in one matrix is exactly equal to the corresponding element in the other matrix.
In the context of our exercise, we equate:

• The first row, first column on both sides, giving us the equation \(3x = 6\).
• The first row, second column, resulting in the equation \(3y = -9\).

This approach of equating matrices allows us to "break down" complex matrix equations into simple linear equations. By solving these smaller, more manageable equations, we can eventually solve for the variables involved. Equating matrices is integral to many areas in math and science, where data dependencies must be aligned and verified.