# Famous Theorems of Mathematics/Algebra/Matrix Theory

An m×n matrix M is a function where A = {1,2...m} × {1,2...n} and F is the field under consideration.

An m×n matrix (read as m by n matrix), is usually written as:

For other related definitions please see this link.

## Basic Proofs

edit1. *The set of all m×n matrices forms an abelian group under matrix addition.*

*Proof*: Clearly the sum of two m×n matrices is another m×n matrix. If A and B are two matrices of equal order then working with their (i,j)th entries we have which proves A+B = B+A i.e. commutativity. For associativity we proceed similarly so that A + (B + C) = (A + B) + C. Also the m×n matrix with all entries zero is the additive identity. For every matrix A, the matrix -A whose (i,j)th entry is is the inverse. So matrices of same order form an abelian group under addition.

2. *Scalar Multiplication has the following properties:*

- 1. Left distributivity: (α+β)A = αA+βA.
- 2. Right distributivity: α(A+B) = αA+αB.
- 3. Associativity: (αβ)A=α(βA)).
- 4. 1A = A.
- 5. 0A=
**0**. - 6. (-1)A = -A.

- (0,1,-1,α & β are scalars; A & B are matrices of equal order,
**0**is the zero matrix.)

- (0,1,-1,α & β are scalars; A & B are matrices of equal order,

*Proof*: Start with the left hand side of (1). We will work with the (i,j)th entries. Clearly and so (1) is proved. Similarly (2) can be proved. Associativity follows as . (4), (5) and (6) follow directly from the definition.

3. *Matrix multiplication has the following properties:*

- 1. Associativity: A(BC) = (AB)C.
- 2. Left distributivity: A(B+C) = AB+AC.
- 3. Right distributivity: (A+B)C = AC+BC.
- 4. IA = A = AI.
- 5. α(BC) = (αB)C = B(αC).

- (α is a scalar; A, B & C are matrices, I is the identity matrix. A,B,C & I are of orders m×n, n×p, p×r & m×m respectively.)

*Proof*: We work with the (i,j)th entries and prove (1) only. The proofs for the rest are similar. Now and also so that (i,j)th entries on the two sides are equal.

4. *Let A and B be m×n matrices. Then:*

- (i) =
- (ii)
- (iii)

*Sketch of Proof*: Work with the (i,j) entries as in the previous proofs.

5. *Any system of linear equations has either no solution, exactly one solution or infinitely many solutions.*

*Proof:*Suppose a linear system Ax = b has two different solutions given by X and Y. Then let Z = X - Y. Clearly Z is non zero and A(X + kZ) = AX + kAZ = b + k(AX - AY) = b + k(b - b) = b so that X + kZ is a solution to the system for every possible value of k. Since k can assume infinitely many values so clearly we have an infinite number of solutions.

6. *Any triangular matrix A satisfying is a diagonal matrix.*

*Proof*: Suppose A is lower triangular. Now the (i,i)th entry of is given by . Also the (i,i)th entry of is given by . Now as so and as can be subtracted from the two sides we are left with .

Now if i = 1 then we have which gives us . Similarly for i =2 we have so that . It is now clear that in this fashion all non diagonal entries of A can be shown to be zero. The proof for an upper triangular matrix is similar.