MATH 323 Theorems
For all and for all , the indicated operations are defined:
- (commutativity of addition)
- (associativity of addition)
- (associativity of multiplication)
- (right distributivity)
- (left distributivity)
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
Equivlent Conditions for Nonsingularity
Let be a matrix. Then the following are equivalent:
- is nonsingular
- has only trivial solution
- 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.
The determinant can be expressed as a cofactor expansion using any row or column of :
If is a matrix, then
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 .
For a matrix
- If has a row or column consisting entirely of 0s, then
- If has two identical rows or two identical columns, then
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.
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 .