Understanding and Solving the Equation ( A^2 3A ) for a 3x3 Nonsingular Matrix

Introduction

The equation ( A^2 3A ) represents a fundamental relationship between a matrix ( A ) and is a common problem in linear algebra. Here, we explore the implications of this equation for a 3x3 nonsingular square matrix, providing a step-by-step solution and demonstrating the importance of the properties of matrices and their determinants.

Given Equations and Properties

Consider the equation A2 3A, where A is a 3x3 nonsingular square matrix. A nonsingular matrix, also known as an invertible matrix, has a non-zero determinant and no zero eigenvalues. In this case, we recognize that ( A ) is a nonsingular matrix, meaning ( A eq 0 ).

Step-by-Step Solution

1. Rearranging the Equation: Start by rearranging the given equation:

[ A^2 - 3A 0 ]

2. Factoring the Equation: Factor out ( A ) from the equation:

[ A(A - 3I) 0 ]

Here, ( I ) represents the 3x3 identity matrix. Since ( A ) is nonsingular, it implies that ( A - 3I eq 0 ). Therefore, the only solution is:

[ A - 3I 0 Rightarrow A 3I ]

Determinant of ( A )

Next, we need to find the determinant of ( A ). Given that ( A 3I ), the determinant can be calculated as follows:

[ det(A) 3^3 cdot det(I) ]

Since the determinant of an identity matrix is 1, we have:

[ det(A) 3^3 cdot 1 27 ]

Conclusion

The value of ( A ) is ( 27 ). Therefore, the solution to the problem is:

[ boxed{27} ]

Additional Considerations

Multiplying the given equation ( A^2 3A ) by taking the determinant on both sides, we get:

[ det(A^2) det(3A) Rightarrow (det(A))^2 3^3 cdot det(A) Rightarrow (det(A))^2 27 cdot det(A) ]

This simplifies to:

[ det(A)(det(A) - 27) 0 ]

Since ( A ) is nonsingular, ( det(A) eq 0 ), leaving us with:

[ det(A) 27 ]

Final Answer

The step-by-step reasoning and calculations consistently lead us to the conclusion that the determinant of ( A ) is 27. Thus, the value of ( A ) is:

[ boxed{27} ]