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 solution to the given matrix equation is \(x = -1\), \(y = -4\), and \(z = -4\).

Step 1: Compare corresponding matrix elements

Since the matrices are equal, their corresponding elements must also be equal. So we have: \[x - 1 = -2\] \[y + 3 = -1\] \[z + 2 = -2\]

Step 2: Solve for x

We can solve the first equation for x by adding 1 to both sides: \[x = -1\]

Step 3: Solve for y

Similarly, we can solve the second equation for y by subtracting 3 from both sides: \[y = -4\]

Step 4: Solve for z

Finally, we can solve the third equation for z by subtracting 2 from both sides: \[z = -4\]

Step 5: Present the solution

Now that we have solved for all the variables, we can present the solution as: \[x = -1\] \[y = -4\] \[z = -4\]

Key Concepts

Matrix Equality

Matrix equality refers to the concept of two matrices being identical. This means that every corresponding element in both matrices must be equal. Imagine two grids filled with numbers, one to the left and the other to the right. For these grids (or matrices) to be the same, each cell must mirror the other in both position and value.

• Each element in one matrix has an equal counterpart in the other matrix.
• The matrices must have the same dimensions (same number of rows and columns).

If the matrices don’t adhere to these conditions, they are not equal. In our exercise, the matrices are given as being equal, allowing us to equate each corresponding element to create solvable equations for variables.

Solving Equations

To solve equations within the context of matrix equality, we first break down each equation derived from corresponding elements of the matrices. It’s like finding the missing puzzle pieces when setting both matrices side by side. Here's how we do it:

• Identify which elements have variables like \(x\), \(y\), or \(z\) that need solving.
• Use simple algebraic operations to solve for these variables.

Take the first equation in our example: \(x - 1 = -2\). To solve for \(x\), we simply add 1 to both sides, simplifying the equation to find \(x = -1\). By following similar steps, we solve for the rest of the variables. Remember, solving these equations often just involves basic arithmetic and maintaining equality at each step.

Linear Algebra Concepts

Understanding these concepts makes matrices much less daunting in algebra. Linear algebra techniques help simplify complex structures like matrices into outputs we can understand and work with. Here are some core ideas:

• Matrix Order: Reflects the number of rows and columns, crucial for comparison.
• Element Position: Each element has a unique position defined by its row and column.
• Variable Representation: Matrices can include variables, making them flexible tools for constructing equations.

With these basic concepts in hand, solving matrix-related problems starts feeling more straightforward. The exercise illustrates how these principles come together: we use the fixed structure of matrices (order and element position) to build equations that we can solve using basic arithmetic.