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Suppose $A$ is the $3 \times 3$ zero matrix (with all zero entries). Describe the solution set of the equation $A \mathbf{x}=\mathbf{0} .$

The solution set is all vectors in $R^{3}$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

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in this example we're dealing with the three by 30 matrix A that's provided here for such a matrix. We can ask interesting questions such as if we were to solve a matrix equation eight times X equals, say, zero vector. What would be the solution set to such a system? Well, this is kind of a strange deal, since our Matrix A has nothing but zero entries. So if we go to the augmented matrix for this system, then it would have this form. First. I'm inserting a as green. So there's a three by 30 matrix and I'll use blue for the augmented portion. So there it is. This is the augmented matrix for eight times X equals zero. Interestingly, the variables are x one x two x three, and we have that x one x two x three are all free variables, since if a is zero matrix, there is just no pivot columns, and that makes all variables free in this situation. Well, in this case, if we were to describe the solution sets to eight times X equals zero matrix, the solution would be X, which is equal to x one x two x three and if we write this in Parametric Victor form, it takes on the form of X one times 100 plus x two times 010 plus x three The last three free variable time +001 where x one x two and x three are in our the set of real numbers. So this is the solution set in Parametric Vector form. But possibly this is just a descriptive f x one x two x three could be any real number whatsoever. And this is the solution, then the solution set to our matrix equation is Theo. Entire set are three. So that's the solution. Sets to a X equals zero when we're dealing with the zero matrix.

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