Abstract
This paper strives to prove different invertible matrices theorems.
The following are some of the theorems that the paper will justify
- A is an invertible matrix
- A is row equivalent to the n × n identity matrix
- A has n pivot positions
- The equation Ax=0 has only the trivial solution
- The equation Ax=b has at least one solution for each b in RN.
- The columns of A spans RN
- The linear transformation X→Ax maps RN onto RN.
- There is an n × n matrix C such that AC=I.
- There is an n × n matrix D such that AD=I.
- The columns of A form a basis of RN.
Keyword: a↔b this means a implies b.
Invertible Matrix Theorems
- Definition of Logically equivalent: Two statements are said to be logically equivalent if and only if they are true in precisely the same situation. Mathematically A and B are logically equivalent if one can be proved using the other that is in notation A ⇔ B (Epp, 2011, p. 30).
- Provide an interpretation for the given statement: The n × n matrix A is invertible: n × n matrix A is said to be invertible if there exist a matrix B with the property that AB=BA=I where I is the identity matrix.
Also the” matrix is of order n × n that is there are n number of rows and n number of columns or it is a square matrix of order n. Invertible implies that an inverse exists of the matrix or the value of the determinant of the matrix is a non-zero number” states Kyle.
We can explain this by an example
A= 2 5
1 3 where n=2
A-1=1/|A|
|A|= (2*3)-(5*1) =6-5=1
Hence there exist an inverse A-1 of A
- Justify that the ten statements are logically equivalent to the statement: The n x n matrix A is invertible.
- Two matrices are said to be row equivalent if and only if they have the same row space that is R (A) = R (B). The row space of the In is the Rn and the row of the An × n are independent, so we can clearly see that R(An × n) = An × n thus a↔b
- Matrix An × n and the identity matrix In is augmented together as (An × n In) and some row operations are done on the matrix to reduce An × n into identity matrix and since An × n is invertible, the final matrix will be (InA-1).since this operation is justified, An × n must have n pivot position hence a↔c.
- If A-1 exist, we have a Ax=A0 implying x=0 and these is the only trivial solution hence a↔d
- If Ax=b has one solution for every b in Rn, then the matrix A must be invertible by the invertible matrix theorem. That is Ax= Ab implies x=Ab. Thus the equation Ax = b has at least one solution for each b in Rn hence a↔e.
- If A is invertible, then A-transposed is invertible. Therefore, the rows of A-transposed span Rn, so the columns of A span Rn. We can also say that if the columns of A spans Rn the same as saying that Ax=b has a solution for ever b in Rn, but if Ax=0 has only the trivial solution, then there are no free variables, so every column of A has a pivot, so Ax=b can never have a pivot in the augmented column. Thus Ax=b has a solution no matter what b namely x=A-1 b hence the column of A span Rn that is a↔f (Zhang, 2011, p. 97)
Invertible Matrix Theorems
- (e), (f), and (g) are equivalent for any matrix, (For a particular A, all statements must be all true or all false. C: The columns of A span RM Let T: RN-->RM be a linear transformation and let A be the standard matrix for T. A: T maps RN onto RM if and only if the columns of A span RM). Thus, (f) implies (g) because they are linked to (e) which is linked to (a).
- If A is an invertible matrix then there is an n × n matrix C such that CA=I: C is the inverse of A and a matrix multiplied by it's inverse is the identity.a↔h
- If an n × n matrix A is invertible, then the columns of AT are linearly independent, that is matrix A must have both rows and columns that are independent for it to be invertible. If A is an invertible matrix then there exists an n × n matrix D such that DA=I. If D equals the inverse of A, then DA will equal the identity.a↔i
- We have already shown that columns of A are independent and they forma span of Rn. Therefore they form a basis for Rn. Thus a ⇔ j.
References
Epp, S. S. (2011). Discrete mathematics with applications. Boston, MA: Brooks/Cole.
kyle. (n.d.). Provide an interpretation for the given statement: The n × n matrix a is invertible?. Retrieved from http://answers.yahoo.com/question/index?qid=20130317064114AAKfVKX
Zhang, F. (2011). Matrix theory: Basic results and techniques. New York [etc..: Springer.
