Problem 40 - Algebra and Trigonometry

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, y = -3.

Step 1: Understand the Equation

We are given a matrix equation where the matrix on the left is multiplied by 3 and equated to another matrix: \[ 3 \begin{bmatrix} x & y \ y & x \end{bmatrix} = \begin{bmatrix} 6 & -9 \ -9 & 6 \end{bmatrix} \]. Our goal is to find the values of \(x\) and \(y\).

Step 2: Distribute the Scalar Multiplication

Multiply the scalar 3 through the matrix on the left-hand side: \[ \begin{bmatrix} 3x & 3y \ 3y & 3x \end{bmatrix} \]. The equation now looks like this: \[ \begin{bmatrix} 3x & 3y \ 3y & 3x \end{bmatrix} = \begin{bmatrix} 6 & -9 \ -9 & 6 \end{bmatrix} \].

Step 3: Set Up the Equations by Comparing Matrices

Equate corresponding elements from the matrices: 1. From the first row and first column: \( 3x = 6 \).2. From the first row and second column: \( 3y = -9 \).3. From the second row and first column: \( 3y = -9 \).4. From the second row and second column: \( 3x = 6 \).

Step 4: Solve for \(x\)

From the equation \( 3x = 6 \), divide both sides by 3 to solve for \(x\): \[ x = \frac{6}{3} \]. Thus, \( x = 2 \).

Step 5: Solve for \(y\)

From the equation \( 3y = -9 \), divide both sides by 3 to solve for \(y\): \[ y = \frac{-9}{3} \]. Thus, \( y = -3 \).

Step 6: Verify Solutions

We check that \(x = 2\) and \(y = -3\) satisfy all equations found in Step 3:1. \(3x = 6\), gives \(3(2) = 6\), true.2. \(3y = -9\), gives \(3(-3) = -9\), true.Thus, both solutions satisfy the original matrix equation.

Key Concepts

Matrix Equation

A matrix equation is very similar to an algebraic equation, but instead of dealing with numbers or variables individually, you deal with matrices as whole entities. In our exercise, we are given a matrix equation that looks like this:

• The left-hand side has a scalar multiplication of a matrix by 3, and
• The right-hand side features another fully formed matrix.

Our task becomes to find values for the variables inside the left-hand side matrix such that when multiplied by the scalar number, both sides become equal. Consider a matrix as a group of numbers arranged in rows and columns. So dealing with matrix equations means you need to consider each number in its position relative to the entire equation. Understanding this layout is fundamental for solving the equations effectively.

Scalar Multiplication

Scalar multiplication involves multiplying each entry of a given matrix by a single number, known as a scalar. This operation is crucial when solving matrix equations as outlined in our problem. Let's see how it works:
The original equation includes a scalar (3) and a matrix containing variables: \[ 3 \begin{bmatrix} x & y \ y & x \end{bmatrix} \] When you distribute the scalar across the matrix, it interacts with each element. Therefore:

• Each 'x' becomes '3x'
• Each 'y' becomes '3y'

This step is vital to simplify the matrix equation. Without it, it would be impossible to equate and solve for the unknown variables in each matrix position.

Matrix Comparison

Once the scalar multiplication is performed, you end up with two matrices that need to be compared. The crucial step here is to look at each element in the matrices as being part of a pair where:

• The first element in the first row and column on the left must equal the first element in the first row and column on the right, and so on.

Basically, you're setting up individual equations based on these comparisons:

• For example, matching the first row, first column from both matrices gives us the equation \(3x = 6\).
• Similarly, matching the first row, second column results in \(3y = -9\).

Each element in the position must agree across both matrices for the equation to hold true.

Solve for Variables

Finally, solving for the variables is like solving for "x" in simple algebra, once you've set up equations through matrix comparisons. We take each of these comparison equations and isolate the variable:

• For the equation \(3x = 6\), simply divide both sides by 3 to solve for \(x\). This gives \(x = 2\).
• Similarly, with \(3y = -9\), dividing by 3 gives \(y = -3\).

It's important to verify your solutions by substituting them back into the original matrix equation. This not only ensures the accuracy of your solution but also confirms that both sides of the matrix equation balance perfectly.