Understanding the Trivial Solution of the Equation A^3x = 0 in Systems of Linear Equations

If A is an n × n matrix for which the equation Ax = b has at least one solutions for every b ∈ R n, then the equation A3x = 0 has only the trivial solution. (T/F)

The statement is True

The statement is True.

To understand why, let’s break down the given information and analyze it.

Firstly, the equation Ax = b represents a system of linear equations, where A is an n x n matrix, x is a vector of unknowns, and b is a vector of constants. The statement states that this system has at least one solution for every b in R^n. In other words, no matter what values we choose for b, we can always find a solution for the system.

Now, consider the equation A^3x = 0. This represents the matrix A raised to the power of 3 (A^3) multiplied by vector x, which results in the zero vector.

If we assume that Ax = b has at least one solution for every b in R^n, it implies that the matrix A is full rank, meaning it has a rank equal to n. This is because if A were not full rank, there would be certain values of b for which the system Ax = b would not have a solution. Since the statement assumes that every b has a solution, it implies that A is full rank.

Now, let’s consider A^3. If A is full rank, then A^3 will also be full rank. This means that A^3 is invertible, and its null space only contains the zero vector. In other words, there are no non-trivial solutions to A^3x = 0.

Therefore, since A^3x = 0 only has the trivial solution (x = 0), the statement “the equation A3x = 0 has only the trivial solution” is indeed true.

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