Mastering 12th Business Maths Exercise 6.1: A Step-by-Step Guide to Solutions

Mastering 12th Business Maths Exercise 6.1: A Step-by-Step Guide to Solutions

As a student studying Business Mathematics, you might have come across Exercise 6.1. This exercise can be quite complex, but with the right approach, you can master it with ease. In this article, we’ll guide you step-by-step through the solutions to this exercise and help you understand the concepts along the way.

Understanding Exercise 6.1

Exercise 6.1 involves the application of matrices in solving different types of problems. The exercise covers topics such as determinants, inverses, eigenvalues, eigenvectors, and more. To solve the exercise, you need to have a good understanding of these concepts and know how to apply them to solve problems efficiently.

Step-by-Step Guide to Solutions

Here’s a step-by-step guide to solving Exercise 6.1:

Step 1: Understand the Problem

The first step to solving any problem is to understand it. Read the problem carefully, and try to identify what you need to solve and the given values. This will help you determine which approach to use in solving the problem.

Step 2: Use Matrices to Represent the Problem

Once you’ve identified the problem’s key features, use matrices to represent it. This will make it easier to solve the problem using matrix operations such as addition, subtraction, multiplication, and division.

Step 3: Apply the Appropriate Matrix Operation

Apply the appropriate matrix operation to the matrices to solve the problem. If you’re finding determinants, you can use properties like row and column expansions, which simplify the process. Similarly, for finding inverses, you can use the adjoint and determinant formula. For eigenvalues and eigenvectors, you need to apply the characteristic equation.

Step 4: Check Your Results

After solving the problem, double-check your results to ensure they’re correct. You can do this by re-applying the matrix operations to your results to see if they match the original values.

Examples and Case Studies

Let’s take a look at some examples of Exercise 6.1 and how to solve them using matrices:

Example 1: Finding the Inverse of a 2×2 Matrix

Suppose you’re given a 2×2 matrix A = [4 5, 2 3]. To find the inverse of the matrix, you can use the adjoint and determinant formula. The determinant of A is (4*3 – 5*2) = 2, so the adjoint matrix is [3 -5, -2 4]. The inverse of matrix A is therefore (1/2) * [3 -5, -2 4] = [3/2 -5/2, -1 2].

Example 2: Using Eigenvectors to Find Eigenvalues

Suppose you’re given a matrix A = [5 2, 2 5]. To find the eigenvalues and eigenvectors of the matrix, you can start by finding the characteristic equation det(A – λI) = (5-λ)*(5-λ) – 4 = λ² – 10λ + 21 = (λ-7)*(λ-3). The eigenvalues are λ1 = 7 and λ2 = 3. To find the eigenvectors, you can substitute each eigenvalue back into the equation (A-λI)x = 0 and solve for x.

Key Takeaways

Exercise 6.1 can be challenging, but with the right approach, you can master it quickly. To do this, you need to understand the problem, use matrices to represent it, apply the appropriate matrix operation, and check your results. Remember to practice regularly and use relevant examples and case studies to solidify your understanding of the concepts. Good luck!