Understanding the Nontrivial Solution of the Equation Ax = 0 in Linear Algebra

If A is 3×3 with three pivot positions, then Ax = 0 has a nontrivial solution.

To determine whether the equation Ax = 0 has a nontrivial solution, we need to examine the rank of matrix A and the number of columns it has

To determine whether the equation Ax = 0 has a nontrivial solution, we need to examine the rank of matrix A and the number of columns it has.

Given that matrix A is 3×3 and has three pivot positions, it means that all three columns of A are linearly independent. The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In this case, since all three columns are linearly independent, the rank of A is 3.

Now, the equation Ax = 0 represents a homogeneous system of linear equations. It essentially means that we are looking for a non-zero vector x such that when multiplied by A, the result is the zero vector.

From basic linear algebra, we know that for a homogeneous system of linear equations to have a nontrivial solution (a solution with non-zero values for the variables), the number of variables (columns) must exceed the rank of the coefficient matrix. In this case, we have 3 variables (x, y, z) and the rank of A is also 3. This tells us that the system has the possibility of having a nontrivial solution.

Therefore, based on the given information, we can conclude that the equation Ax = 0 does indeed have a nontrivial solution.

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