Question

Let \(A\) and \(B\) be \(n \times n\) matrices for which the systems of equations \(A \mathbf{x}=\mathbf{0}\) and \(B \mathbf{x}=\mathbf{0}\) each have only the trivial solution \(\mathbf{x}=\mathbf{0}\). Show that the system \((A B) \mathbf{x}=\mathbf{0}\) has only the trivial solution.

Short answer

The system \((AB)\mathbf{x} = \mathbf{0}\) also has only the trivial solution \(\mathbf{x} = \mathbf{0}\).

Step-by-step solution

Understand the Given Conditions

We are given two matrices, \(A\) and \(B\), both of \(n \times n\) dimensions. The condition that the systems \(A \mathbf{x} = \mathbf{0}\) and \(B \mathbf{x} = \mathbf{0}\) only have trivial solution \(\mathbf{x} = \mathbf{0}\) implies that both \(A\) and \(B\) are invertible matrices.

Consider the System \\( (AB)\mathbf{x} = \mathbf{0} \\\

We need to show that the system \((AB) \mathbf{x} = \mathbf{0}\) also has only the trivial solution \(\mathbf{x} = \mathbf{0}\). Because both \(A\) and \(B\) are invertible, \(AB\) is also invertible.

Use the Invertibility of the Product

Since \(AB\) is invertible, the matrix \(AB\) also has full rank, meaning the linear equation system \((AB)\mathbf{x} = \mathbf{0}\) must have a unique solution. As it specifically has \(\mathbf{x} = \mathbf{0}\) as a solution, being \(AB\) invertible confirms that the trivial solution is indeed the only solution.

Conclusion

Because \((AB)\mathbf{x} = \mathbf{0}\) has a unique solution and that solution is trivial, we conclude that this system, like the others, has only the solution \(\mathbf{x} = \mathbf{0}\). Therefore, \( (AB) \mathbf{x} = \mathbf{0} \) has only the trivial solution.

Key concepts

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices to produce a third matrix. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.
In this exercise, we deal with square matrices of order \(n\), meaning the matrices have \(n\) rows and \(n\) columns.

• When multiplying matrices, remember that order matters. \(AB\) is not necessarily equal to \(BA\).
• The resulting matrix \(C = AB\) in the context of square matrices will also be an \(n \times n\) matrix.

Matrix multiplication involves combining the corresponding elements of the rows of the first matrix and the columns of the second matrix using the dot product.
This operation results in a new matrix that encompasses the applied linear transformations from both original matrices.In applications like solving systems of linear equations, matrix multiplication allows us to transform and simplify problems. It is inherently tied to understanding how linear equations can be represented and manipulated.

Linear Equations

Linear equations are equations of the first degree, forming straight lines when graphed. In matrix terms, systems of linear equations can be represented in the form \(A \mathbf{x} = \mathbf{b}\). Here, \(A\) is a matrix of coefficients, \(\mathbf{x}\) is a vector of variables, and \(\mathbf{b}\) is a vector of constants.
- When \(\mathbf{b} = \mathbf{0}\), the system is homogeneous.- Solutions to these equations involve finding vectors \(\mathbf{x}\) that satisfy the matrix equation.For a homogeneous system \(A \mathbf{x} = \mathbf{0}\), having only the trivial solution \(\mathbf{x} = \mathbf{0}\) suggests that the matrix \(A\) is invertible. Invertible matrices are non-singular and possess certain key characteristics: they have a determinant other than zero and a full rank equal to their number of rows and columns.
In solving such systems, especially in mathematical and engineering applications, understanding linear equations is essential, as it helps find vectors that align with constraints dictated by the given matrix.

Trivial Solution

In the context of systems of linear equations, a trivial solution refers to the solution where all variables are zero. When dealing with homogeneous systems of the form \(A \mathbf{x} = \mathbf{0}\), having only the trivial solution means that \( \mathbf{x} \), the vector in the equation, can only be the zero vector.
This situation is often due to the matrix being invertible, ensuring there are no free variables that allow other solutions. An invertible matrix in this context guarantees that there's a unique solution.
- If both matrices \(A\) and \(B\) ensure their respective systems only yield the trivial solution, their product \(AB\) will also ensure the system \((AB) \mathbf{x} = \mathbf{0}\) is trivial.- Thus, the importance of having invertible matrices becomes clear, as they determine the ability to only allow the zero vector as a solution, leading to more predictable and stable outcomes in mathematical equations and modeling.Understanding the trivial solution in-depth aids in comprehending how adjustability within a system affects the complexity of solution spaces and how constraints are defined.