Solving Matrix Equations Find P Q R And S

In this comprehensive guide, we will delve into the fascinating world of matrix equations, specifically focusing on solving for unknown variables within matrix operations. Matrix equations are fundamental in various fields, including linear algebra, computer graphics, and engineering. Understanding how to manipulate and solve these equations is crucial for anyone working with mathematical models and data analysis.

Understanding Matrix Equations

Matrix equations are mathematical expressions that involve matrices, which are rectangular arrays of numbers arranged in rows and columns. These equations often contain unknown variables that we aim to determine. The beauty of matrix equations lies in their ability to represent complex systems of linear equations in a compact and elegant form. They also provide a powerful framework for solving systems of equations with multiple variables.

Before diving into solving matrix equations, it is essential to grasp the basic operations involving matrices. These include matrix addition, subtraction, and multiplication. Matrix addition and subtraction are performed element-wise, meaning that corresponding elements in the matrices are added or subtracted. Matrix multiplication, on the other hand, follows a more intricate rule, where the elements of the resulting matrix are obtained by taking the dot product of rows from the first matrix and columns from the second matrix.

Deconstructing the Matrix Equation

Let's consider the given matrix equation:

$\left[\begin{array}{cc}2 p-1 & 2 q \ 1 & r-s\end{array}\right]+\left[\begin{array}{cc}1 & 2-q \ 3 s & 2 r-1\end{array}\right]=\left[\begin{array}{cc}3 p & 4 \ 2 s-r & 4\end{array}\right]+\left[\begin{array}{cc}4 & 3 q-1 \ 2 & p+s\end{array}\right]$

This equation involves two matrices on each side, connected by the addition operation. Our goal is to find the values of the unknowns p, q, r, and s that satisfy this equation. To achieve this, we will utilize the properties of matrix addition and equality.

Matrix equality implies that two matrices are equal if and only if their corresponding elements are equal. In other words, the element in the i-th row and j-th column of the first matrix must be equal to the element in the i-th row and j-th column of the second matrix for all i and j. This principle is the cornerstone of solving matrix equations.

Solving for Unknowns

To solve for the unknowns p, q, r, and s, we will perform matrix addition on both sides of the equation and then equate the corresponding elements. Let's start by adding the matrices on each side:

Left-hand side:

$\left[\begin{array}{cc}2 p-1 & 2 q \ 1 & r-s\end{array}\right]+\left[\begin{array}{cc}1 & 2-q \ 3 s & 2 r-1\end{array}\right] = \left[\begin{array}{cc}(2p - 1) + 1 & 2q + (2 - q) \ 1 + 3s & (r - s) + (2r - 1)\end{array}\right] = \left[\begin{array}{cc}2p & q + 2 \ 1 + 3s & 3r - s - 1\end{array}\right]$

Right-hand side:

$\left[\begin{array}{cc}3 p & 4 \ 2 s-r & 4\end{array}\right]+\left[\begin{array}{cc}4 & 3 q-1 \ 2 & p+s\end{array}\right] = \left[\begin{array}{cc}3p + 4 & 4 + (3q - 1) \ (2s - r) + 2 & 4 + (p + s)\end{array}\right] = \left[\begin{array}{cc}3p + 4 & 3q + 3 \ 2s - r + 2 & p + s + 4\end{array}\right]$

Now, we have the simplified matrix equation:

$\left[\begin{array}{cc}2p & q + 2 \ 1 + 3s & 3r - s - 1\end{array}\right] = \left[\begin{array}{cc}3p + 4 & 3q + 3 \ 2s - r + 2 & p + s + 4\end{array}\right]$

By equating the corresponding elements, we obtain a system of four linear equations:

1. 2p = 3p + 4
2. q + 2 = 3q + 3
3. 1 + 3s = 2s - r + 2
4. 3r - s - 1 = p + s + 4

Now, we can solve this system of equations to find the values of p, q, r, and s. Let's solve each equation one by one:

Solving for p

From the first equation, 2p = 3p + 4, we can isolate p:

2p - 3p = 4

-p = 4

p = -4

Solving for q

From the second equation, q + 2 = 3q + 3, we can isolate q:

q - 3q = 3 - 2

-2q = 1

q = -1/2

Solving for r and s

We now have two equations with two unknowns, r and s:

1. 1 + 3s = 2s - r + 2
2. 3r - s - 1 = p + s + 4

Substitute the value of p = -4 into the fourth equation:

3r - s - 1 = -4 + s + 4

3r - s - 1 = s

3r = 2s + 1

Now, let's simplify the third equation:

1 + 3s = 2s - r + 2

s + r = 1

r = 1 - s

Substitute r = 1 - s into the equation 3r = 2s + 1:

3(1 - s) = 2s + 1

3 - 3s = 2s + 1

5s = 2

s = 2/5

Now, substitute s = 2/5 back into r = 1 - s:

r = 1 - 2/5

r = 3/5

Therefore, the solution to the matrix equation is:

• p* = -4
• q* = -1/2
• r* = 3/5
• s* = 2/5

Verification

To ensure the accuracy of our solution, we can substitute these values back into the original matrix equation and verify that both sides are equal. This step is crucial in confirming that we have found the correct solution.

Conclusion

Solving matrix equations involves manipulating matrices and their elements to isolate and determine the values of unknown variables. By understanding matrix operations, equality conditions, and systematic equation-solving techniques, you can confidently tackle a wide range of matrix equations. This skill is essential for various applications in mathematics, science, and engineering.

This comprehensive guide has provided you with a solid foundation for solving matrix equations. Remember to practice and apply these techniques to different types of problems to further enhance your understanding and proficiency. Matrix equations are a powerful tool, and mastering them will open doors to solving complex problems in various fields.