Suppose a is n x n matrix and the equation ax = 0 has only the trivial solution. explain why a has n pivot columns and a is row equivalent to in. by theorem 7, this shows that a must be invertible.)
theorem 7: an n x n matrix a is invertible if and only if a is row equivalent to in, and in this case, any sequence of elementary row operations that reduces a to in also transfrms in into a-1.

Remember, a homogeneous system always is consistent. Then we can reason with the rank of the matrix.

If the system Ax=0 has only the trivial solution that's mean that the echelon form of A hasn't free variables, therefore each column of the matrix has a pivot.

Since each column has a pivot then we can form the reduced echelon form of the A, and leave each pivot as 1 and the others components of the column will be zero. This means that the reduced echelon form of A is the identity matrix and so on A is row equivalent to identity matrix.

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