Proving the Relationship Between the Trivial Solution and Row Equivalence to the Identity Matrix

If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.

To prove that if the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix, we need to demonstrate two key facts:

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To prove that if the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix, we need to demonstrate two key facts:

1. If the equation Ax = 0 has only the trivial solution, then the only solution to the system of linear equations represented by the augmented matrix [A | 0] is x = 0.

2. If the only solution to the system of linear equations represented by the augmented matrix [A | 0] is x = 0, then A is row equivalent to the n x n identity matrix.

Proof:

Let’s start by assuming that the equation Ax = 0 has only the trivial solution. This means that for the given matrix A, there are no nonzero vectors x that satisfy the equation Ax = 0, except for x = 0 (the trivial solution).

Now, let’s represent the system of linear equations using the augmented matrix [A | 0]. The number of columns in A is n since A is an n x n matrix, and there is an additional column for the right-hand side (0 vector).

Since the equation Ax = 0 has only the trivial solution, this means that there is no nonzero vector x that satisfies the equation Ax = 0. In other words, the system of linear equations represented by [A | 0] has only the solution x = 0.

This implies that the row reduction of the augmented matrix [A | 0] will lead to a row echelon form that consists of the identity matrix on the left-hand side and the zero vector on the right-hand side.

Now, let’s label the row-reduced form of the augmented matrix as [R | 0], where R represents the row echelon form consisting of the identity matrix and 0’s.

Since the row-reduced form of the augmented matrix [A | 0] is [R | 0], this means that A is row equivalent to the row echelon form matrix R.

To show that A is row equivalent to the n x n identity matrix, we need to demonstrate that R can be transformed into the n x n identity matrix using elementary row operations.

Since R is already in row echelon form and consists of the identity matrix on the left-hand side, it must be an n x n identity matrix.

Therefore, we have established that if the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.

QED

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