MATH 323 Theorems

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MATH 323 Theorems

Chapter 1

Theorem 1.3.2

For all and for all , the indicated operations are defined:

  1. (commutativity of addition)
  2. (associativity of addition)
  3. (associativity of multiplication)
  4. (right distributivity)
  5. (left distributivity)

Theorem 1.3.3

If and are nonsingular matrices, then is also nonsingular and

(note the opposite order on the right-hand side)


For nonsingular matrices , is also nonsingular and

Theorem 1.4.2

Equivlent Conditions for Nonsingularity

Let be a matrix. Then the following are equivalent:

  1. is nonsingular
  2. has only trivial solution
  3. is row equivalent to identity matrix of size ()


(1) → (2): Let be solution of . So can become . , so

(2) → (3): Rewrite in row echelon form as . If one diagonal entry of is 0, then there is at least one free variable and infinitely many solutions, one of which is nonzero. This leads to a contradiction, so all diagonal entries of must be 1, so rref will be identity matrix.

(3) → (1): If is row equivalent to , then there exists a sequence of elementary matrices such that . Therefore, (note reverse order), so is nonsingular.



The system of equations has a unique solution iff is nonsingular.

  • is nonsingular, can be rewritten as
  • Assume unique solution exists. If were singular then homogeneous system would have a nonzero solution (by Theorem 1.4.2). If this were the case, ... there would be another solution, which is a contradiction.

Chapter 2

Theorem 2.1.1

The determinant can be expressed as a cofactor expansion using any row or column of :

Theorem 2.1.2

If is a matrix, then

Theorem 2.1.3

If is a triangular matrx, then is equal to the product of the diagonal entries of .


Proof by Induction.

Basis step. determinant of matrix is product of diagonals.

Inductive step. Assuming determinant of matrix is product of diagonals, can be formed by adding a nonzero row to the top, and a zero column with nonzero to the . The determinant would just be .


Theorem 2.1.4

For a matrix

  1. If has a row or column consisting entirely of 0s, then
  2. If has two identical rows or two identical columns, then

Theorem x.x.x

A matrix is singular iff .


can be reduced to row echelon form by a finite number of row operations. This means that is an upper triangular matrix whose determinant is . The determinants of elementary matrices will never be zero, so if the determinant of is zero, then at least one row of must be all 0.

Such a matrix cannot be row-equivalent to and therefore cannot have an inverse.


Theorem x.x.x


If is singular, it follows from #Theorem 1.5.2 that is also singular (see exercise 14 of Section 1.5). Therefore if or is singular.

Let be nonsingular, so Therefore, . By separating the elementary matrices from the determinant, we arrive that .