Problem 12 - Advanced Engineering Mathematics

Question

Determine the values of \(x\) and \(y\) for which the matrices are equal. $$ \left(\begin{array}{cc} x^{2} & 1 \\ y & 5 \end{array}\right),\left(\begin{array}{rr} 9 & 1 \\ 4 x & 5 \end{array}\right) $$

Step-by-Step Solution

Answer

The solutions are \((x, y) = (3, 12)\) and \((x, y) = (-3, -12)\).

Step 1: Understanding Matrix Equality

Matrix equality means that corresponding elements of two matrices are equal. In this problem, you need to set corresponding elements of the given matrices equal to each other.

Step 2: Set Corresponding Elements Equal

For the matrices \( \begin{pmatrix} x^2 & 1 \ y & 5 \end{pmatrix} \) and \( \begin{pmatrix} 9 & 1 \ 4x & 5 \end{pmatrix} \), the equality of elements gives: 1. \( x^2 = 9 \) for the first row, first column.2. \( 1 = 1 \) is trivially true.3. \( y = 4x \) for the second row, first column.4. \( 5 = 5 \) is trivially true.

Step 3: Solve Equation for \(x\)

From \( x^2 = 9 \), solve for \(x\):\[ x = \pm 3 \].

Step 4: Solve for \(y\) Using Values of \(x\)

Determine \(y\) by substituting the values of \(x\) from the previous step into the equation \( y = 4x \):- If \( x = 3 \), then \( y = 4(3) = 12 \).- If \( x = -3 \), then \( y = 4(-3) = -12 \).

Step 5: Verify Solutions

Plug these values back into the equations to ensure they satisfy both matrices:- For \( (x, y) = (3, 12) \), \( x^2 = 9 \) and \( y = 4x = 12 \).- For \( (x, y) = (-3, -12) \), \( x^2 = 9 \) and \( y = 4x = -12 \). Both pairs satisfy the matrix equality.

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the operations you can perform on them. A matrix is simply a rectangular arrangement of numbers or symbols spread over rows and columns. Understanding the basics of matrix algebra involves knowing how to add, subtract, and multiply matrices, as well as determining their equality. In a matrix, every element has a specific location identified by its row and column number. For instance, the element in the first row and first column of a matrix is often denoted as \(a_{11}\). When performing operations on matrices, it's crucial to ensure that the matrices are compatible.

• Addition and subtraction require that matrices be of the same dimensions.
• Multiplication can be performed if and only if the number of columns in the first matrix equals the number of rows in the second.

This exercise focuses especially on matrix equality, where our goal is to determine if two matrices are perfectly identical across all corresponding elements.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of the variables that make the equation true. In this matrix equality problem, the given equations arise from comparing corresponding elements of two matrices:\[ \begin{pmatrix} x^2 & 1 \ y & 5 \end{pmatrix} = \begin{pmatrix} 9 & 1 \ 4x & 5 \end{pmatrix}\]Setting the elements equal, we derive the equations \(x^2 = 9\) and \(y = 4x\). The key to solving these equations is to treat each one separately and systematically.

• First, consider the equation \(x^2 = 9\). Solving it using basic algebraic techniques, we find that \(x = 3\) or \(x = -3\).
• Next, we use these values of \(x\) to solve \(y = 4x\). Substituting \(x = 3\), we find \(y = 12\). For \(x = -3\), \(y = -12\).

It is critical while solving these equations to ensure that every step adheres to the mathematical principles, resulting in solutions that satisfy all parts of the original matrix equality.

Element Equality

Element equality in matrices means that every element in one matrix must be exactly equal to the corresponding element in another matrix for the two matrices to be considered equal. In the given problem, the matrices are: \[ \begin{pmatrix} x^2 & 1 \ y & 5 \end{pmatrix} \]\[ \begin{pmatrix} 9 & 1 \ 4x & 5 \end{pmatrix} \]The equality of matrices requires that:

• The elements in the first row, first column: \(x^2 = 9\)
• The elements in the second row, first column: \(y = 4x\)
• The constants \(1 = 1\) and \(5 = 5\) are automatically equal.

Using these conditions, you can solve for variable values that satisfy every pair of corresponding elements. The intriguing part about matrix equality is that even a single discrepancy between the elements can cause the matrices not to be equal. Therefore, thorough checking is important to establish true equality.