The class ends and students make the usual mad dash to the exits. Lines of exiting students quickly form as new students flood in. A student approaches me with a brief question as I desperately try to ready the classroom for the incoming professor.
“Dr. B., why are we only interested in the invertibility of square matrices?” The student asks.
“Because, it doesn’t make sense to talk about the invertibility of matrices that are not square,” I respond.
In the study of linear algebra, matrices are a common conversation point. Matrices stem from a wide range of applications. They can be viewed as linear transformations, that is, they map objects from one space to another space. They may represent a vector equation or a linear system of equations. The latter case may be more familiar from your high school algebra training. In either case, a matrix can be used to store all the relevant information, for which allows for easier manipulation by hand or through a computer.
All matrices have N rows (number of equations) and M columns (number of unknowns). If N equals M then it is called a square matrix. When solving a linear system of equations, say A x = b, a unique solution exists if and only if the matrix is invertible, that is, there exists a matrix C such that A C = C A = I, where I is the identity matrix.
This only makes sense for square matrices. Therefore it is nonsense to talk about the inverse of matrices that are not square (rectangular matrices). It is a contradiction of terms. We simply can not have an invertible 2 by 3 matrix. It simply does not make sense. It is like having a square circle or a married bachelor.
What does this have to do with God? Everything. Consider a common dialogue that I have been privy to:
Skeptic : If God is All-Powerful then all things are possible.
Me : No, God can do any thing but not anything.
Skeptic : Huh?
Me : God can not do what is logically inconsistent. He can not make a married person also a bachelor. He can not make a positive number also be a negative number. He can not make a square also a circle.
Skeptic : Oh, okay, but I still do not think God is All-Powerful. Maybe the reason why there is so much pain and suffering in the world is because God is not strong enough to overcome it. Could God create a stone that even He could not lift?
This conversation seems to make a truly profound claim about an attribute of God, that is, that He can not be All-Powerful. However, the last question by the skeptic is simply a contradiction of terms. It doesn’t make sense.
If God is All-Powerful, that is, He can lift anything, then it does not make sense to talk about something He can not lift! This is equivalent to a student wishing to talk about invertible 2 by 3 matrices; be it their wish or not it simply does not make sense from basic principles.
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