Solving For X And Y In Equal Matrices A Comprehensive Guide
In the realm of linear algebra, matrices stand as fundamental structures, serving as powerful tools for representing and manipulating data. A core concept involves matrix equality, which dictates that two matrices are considered equal if and only if their corresponding elements are identical. This article delves into the intricacies of determining the values of variables within matrices that satisfy this equality condition. Specifically, we will focus on the problem of finding the values of x and y for which two given matrices, A and B, are equal. This exploration will not only solidify your understanding of matrix equality but also enhance your problem-solving skills in linear algebra.
Matrix Equality: A Foundation
Before diving into the specifics of our problem, let's solidify the foundational concept of matrix equality. Two matrices, say A and B, are deemed equal if and only if they meet the following two crucial criteria:
- They must have the same dimensions: This means that the number of rows and columns in matrix A must be identical to the number of rows and columns in matrix B.
- Their corresponding elements must be equal: Every element in matrix A must have an exact counterpart in matrix B at the same position, and their values must be identical.
In mathematical notation, if A = [aᵢⱼ] and B = [bᵢⱼ] are two matrices of the same size (m × n), then A = B if and only if aᵢⱼ = bᵢⱼ for all i and j, where i represents the row number and j represents the column number. Understanding this definition is paramount as it lays the groundwork for solving problems involving matrix equality.
Setting the Stage: Our Matrices
With the concept of matrix equality firmly in mind, let's introduce the matrices we'll be working with. We are given two matrices, A and B, defined as follows:
- A =
[ 2x + 1 3y ] [ 0 y² - 3y ] - B =
[ x + 3 y + 2 ] [ 0 8 ]
Our mission is to find the values of x and y that make these two matrices equal. To achieve this, we will leverage the definition of matrix equality, setting corresponding elements equal to each other and solving the resulting equations. This systematic approach will lead us to the desired values of x and y.
Element-by-Element Comparison: Unveiling the Equations
To find the values of x and y that make matrices A and B equal, we must meticulously compare their corresponding elements. According to the definition of matrix equality, two matrices are equal if and only if their corresponding elements are identical. This principle allows us to set up a system of equations by equating the elements in the same positions within the matrices. Let's break down this process step by step.
Examining the Top-Left Elements
We begin by focusing on the elements in the top-left corner of both matrices. In matrix A, this element is 2x + 1, while in matrix B, it is x + 3. For the matrices to be equal, these elements must be equal. This gives us our first equation:
2x + 1 = x + 3
This equation is a simple linear equation in one variable, x. Solving it will provide us with a potential value for x that satisfies the equality condition for these specific elements. We'll proceed to solve this equation later in this section.
Moving to the Top-Right Elements
Next, we shift our attention to the elements in the top-right corner of the matrices. In matrix A, this element is 3y, while in matrix B, it is y + 2. Again, for the matrices to be equal, these elements must be equal. This leads us to our second equation:
3y = y + 2
Similar to the previous equation, this is another linear equation in one variable, y. Solving this equation will give us a potential value for y that satisfies the equality condition for these elements. We will also solve this equation in the subsequent steps.
Analyzing the Bottom-Left Elements
Now, let's consider the elements in the bottom-left corner of the matrices. In both matrices A and B, this element is 0. This observation is crucial because it immediately tells us that the equality condition is satisfied for these elements, regardless of the values of x and y. In other words, this comparison does not provide us with any additional equations or constraints for x and y. However, it does confirm that our matrices are potentially equal, as this particular pair of corresponding elements matches.
The Critical Bottom-Right Elements
Finally, we turn our attention to the elements in the bottom-right corner of the matrices. In matrix A, this element is y² - 3y, while in matrix B, it is 8. For the matrices to be equal, these elements must be equal, giving us our third equation:
y² - 3y = 8
This equation is a quadratic equation in y, which is a step up in complexity from the previous linear equations. Solving this quadratic equation will potentially give us two values for y that satisfy the equality condition for these elements. It's important to note that we need to check if these solutions are consistent with the solution we obtained from the equation involving the top-right elements. If the values don't align, we need to discard any inconsistent solutions.
The System of Equations: A Summary
In summary, by comparing the corresponding elements of matrices A and B, we have successfully derived a system of equations that must be satisfied for the matrices to be equal. This system consists of the following equations:
- 2x + 1 = x + 3
- 3y = y + 2
- y² - 3y = 8
Our next step is to systematically solve this system of equations to find the values of x and y that make matrices A and B equal. This involves applying algebraic techniques to isolate the variables and determine their values.
Solving for x and y: A Step-by-Step Approach
Having established the system of equations derived from the element-by-element comparison of matrices A and B, our focus now shifts to solving for the unknown variables, x and y. We will tackle each equation systematically, employing algebraic techniques to isolate the variables and determine their values. This methodical approach will ensure we find all possible solutions that satisfy the matrix equality condition.
Isolating x: The First Equation
Let's begin with the first equation, which involves the variable x:
2x + 1 = x + 3
To solve for x, our goal is to isolate x on one side of the equation. We can achieve this by performing the following steps:
-
Subtract x from both sides of the equation: This eliminates x from the right side, giving us:
2x - x + 1 = x - x + 3
x + 1 = 3
-
Subtract 1 from both sides of the equation: This isolates x on the left side, resulting in:
x + 1 - 1 = 3 - 1
x = 2
Therefore, the solution for x from the first equation is x = 2. This means that if the matrices are to be equal, the value of x must be 2. We will keep this solution in mind as we proceed to solve for y.
Unveiling y: The Second Equation
Now, let's turn our attention to the second equation, which involves the variable y:
3y = y + 2
Similar to solving for x, we aim to isolate y on one side of the equation. We can do this through the following steps:
-
Subtract y from both sides of the equation: This eliminates y from the right side, leading to:
3y - y = y - y + 2
2y = 2
-
Divide both sides of the equation by 2: This isolates y on the left side, giving us:
(2y) / 2 = 2 / 2
y = 1
Thus, the solution for y from the second equation is y = 1. This indicates that if the matrices are to be equal, the value of y must be 1. We will now consider the third equation to ensure that this solution is consistent.
Tackling the Quadratic: The Third Equation
Our final equation is a quadratic equation in y:
y² - 3y = 8
To solve a quadratic equation, we typically rearrange it into the standard form, which is ay² + by + c = 0, where a, b, and c are constants. Let's rearrange our equation:
-
Subtract 8 from both sides of the equation: This sets the equation equal to zero, giving us the standard form:
y² - 3y - 8 = 0
Now that we have the equation in standard form, we can solve it using various methods, such as factoring, completing the square, or the quadratic formula. In this case, factoring is not straightforward, so we will use the quadratic formula. The quadratic formula states that for an equation of the form ay² + by + c = 0, the solutions for y are given by:
*y* = (-*b* ± √(*b*² - 4*a*c)) / (2*a*)
In our equation, a = 1, b = -3, and c = -8. Substituting these values into the quadratic formula, we get:
*y* = (-(-3) ± √((-3)² - 4 * 1 * (-8))) / (2 * 1)
*y* = (3 ± √(9 + 32)) / 2
*y* = (3 ± √41) / 2
This gives us two possible solutions for y:
*y*₁ = (3 + √41) / 2
*y*₂ = (3 - √41) / 2
Consistency Check: Reconciling the Solutions
We have obtained three potential values for y: y = 1 from the second equation, and y₁ = (3 + √41) / 2 and y₂ = (3 - √41) / 2 from the third equation. However, for the matrices to be equal, the value of y must be consistent across all equations. This means that the value of y obtained from the second equation (y = 1) must also satisfy the third equation.
Let's check if y = 1 satisfies the third equation:
y² - 3y = 8
Substituting y = 1, we get:
(1)² - 3(1) = 1 - 3 = -2
Since -2 ≠ 8, the value y = 1 does not satisfy the third equation. Therefore, we must discard this solution.
This leaves us with the two solutions from the quadratic equation, y₁ = (3 + √41) / 2 and y₂ = (3 - √41) / 2. Since these are the only potential solutions for y that satisfy the third equation, we must consider them further. However, because we obtained a single, unique solution for y from the second equation (y = 1), and this solution is not consistent with the solutions from the third equation, we conclude that there is no value of y that will make the matrices equal.
Conclusion: The Final Verdict
After a thorough exploration of matrix equality and a systematic approach to solving the equations derived from comparing the elements of matrices A and B, we arrive at a conclusive answer. We have determined that:
- The value of x that satisfies the equality condition is x = 2.
- There is no value of y that will make the matrices equal. The value y = 1, obtained from the equation 3y = y + 2, does not satisfy the quadratic equation y² - 3y = 8. The solutions to the quadratic equation, y₁ = (3 + √41) / 2 and y₂ = (3 - √41) / 2, are not consistent with the solution from the linear equation.
Therefore, the matrices A and B can never be equal, regardless of the value of y, as the equality conditions cannot be simultaneously satisfied for all corresponding elements. This conclusion underscores the importance of consistency when dealing with systems of equations arising from matrix equality. Only solutions that satisfy all equations simultaneously are valid. This problem provides a valuable illustration of how the principles of matrix equality and algebraic problem-solving converge to determine the values of variables within matrices.
