INTRODUCTION

In this theme, we will study the matrix equation Ax = b. In particular, we investigate the solution space of an equation. We consider a nonhomogeneous matrix equation, the corresponding homogeneous matrix equation, and the effect of adding solutions of these equations. Finally, we apply matrix methods to find equilibrium conditions for a colony of bacteria to coexist.

CONCLUSION

Our investigation of the equation Ax = b showed that b is a solution if and only if b is a linear combination of the columns of A. We showed that a homogeneous equation always has a solution, and it has more than one solution whenever there is a free variable. Next, we demonstrated that sums of solutions of a homogeneous equation and scalar multiples of solutions of a homogeneous equation are also solutions of that equation. Further, if one adds a solution of a homogeneous equation to a solution of the corresponding nonhomogeneous equation, one gets a solution of the nonhomogeneous equation (that is, if Ap = b and Av = 0, then A( p+v) = 0).

Finally, we studied a colony of bacteria. We found that, under certain prescribed conditions, 2000 of the first type will always be present, while the second and third types of bacteria will be related by the linear relationship x₂ = 6000-2x₃, where x₂ is the number of the second type of bacteria, x₃ is the number of the third type of bacteria, and x₂ and x₃ are both nonnegative.