Problem 1 - Discrete Mathematics with Applications

Question

Solve the following equations. $$\left[\begin{array}{ccc} x-1 & 2 & 0 \\ 0 & y+3 & 4 \\ -3 & 1 & z+2 \end{array}\right]=\left[\begin{array}{rrr} -2 & 2 & 0 \\ 0 & -1 & 4 \\ -3 & 1 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The short answer for the question "Solve the following equations" is: \(x = -1\), \(y = -4\), and \(z = -4\).

Step 1: Finding the equations for x, y and z

Since two matrices are equal if their corresponding elements are equal, we can equate the elements of the two given matrices and obtain the following equations: 1. \(x-1 = -2\) 2. \(y+3 = -1\) 3. \(z+2 = -2\) Now, we can solve these equations for x, y, and z.

Step 2: Solving the equation for x

The first equation is \(x-1 = -2\). To solve for x, we can add 1 to both sides: \(x = -1\)

Step 3: Solving the equation for y

The second equation is \(y+3 = -1\). To solve for y, we can subtract 3 from both sides: \(y = -1 - 3 = -4\)

Step 4: Solving the equation for z

The third equation is \(z+2 = -2\). To solve for z, we can subtract 2 from both sides: \(z = -2 - 2 = -4\)

Step 5: Final answer

By solving the equations, we found the values of the unknowns: \(x = -1\), \(y = -4\), and \(z = -4\).

Key Concepts

Matrix Equality

Understanding matrix equality is crucial when working with matrices. Two matrices are considered equal if every corresponding element in the matrices is identical. This means that for each element in the first matrix, there must be an equal element in the exact same position in the second matrix.
If we have two matrices:

• Matrix A, where element A_ij refers to the element in the i-th row and j-th column.
• Matrix B, where element B_ij is in the same position.

For A and B to be equal, we must have A_ij = B_ij for all i and j.
In our exercise, the matrix on the left needed to be equal to the matrix on the right. This requirement led us to setups of equations for each corresponding element.

Solving Systems of Equations

Solving systems of equations is a common task in mathematics that can be approached in various ways. Once each element from our matrices is matched, they form simple linear equations. For example, from our exercise, we derived three individual equations by comparing the matrices element-by-element.
Here's a simple method to solve these equations:

• Equation 1: Solve for x in the equation: \(x - 1 = -2\). Adding 1 to both sides gives us \(x = -1\).
• Equation 2: Solve for y in the equation: \(y + 3 = -1\). Subtracting 3 from both sides yields \(y = -4\).
• Equation 3: Solve for z in the equation: \(z + 2 = -2\). Again subtracting 2 gives us \(z = -4\).

Using simple algebraic manipulations by isolating the variable on one side, we can find each unknown. These solutions give us x, y, and z values that satisfy the original matrix equation.

Element-wise Comparison in Matrices

Element-wise comparison is an essential operation when determining whether two matrices are equal. It involves checking each corresponding pair of elements to decide whether they are identical.
Consider a matrix where each element, in sequence, is matched to another matrix. This approach requires breaking down matrices into individual elements and then comparing them directly. Thus, disparities are apparent immediately, especially when solving equations.

• The first element of the first row in one matrix is compared to the first element of the first row in the second matrix.
• Continue this comparison across all rows and columns.

After a thorough element-wise comparison, if all comparisons are equal, it confirms matrix equality. If any elements differ, the matrices are not equal, making it necessary to solve particular inequalities if possible.