Theorems

1)      Let A,B,C, and D be m x n matrices

A)    A+B=B+A  

B)     A+(B+C)=(A+B)+C

C)    There is a unique m x n matrix O such that  A+O = A.  Matrix O is the zero matrix or the additive identity.

D)    There is a unique m x n matrix D such that A+D = 0.  We can write D as –A therefore A+(-A) = 0.  Matrix –A is the additive inverse or negative of A.

2)      Let A,B,C, and D be m x n matrices of the appropriate sizes

a)      A(BC) = (AB)C

b)      A(B+C) = AB+AC

c)      (A+B)C = AC+BC

3)      Let A and B be matrices and  r and s are real numbers

a)      r(sA) = (rs)A

b)      (r+s)A = rA+sA

c)      r(A+B) = rA+sA

d)      A(rB) = r(AB) = (rA)B

4)      Let A and B be matrices and s is a scalar

a)      (A^T)^T =  A

b)       (A+B)^T = A^T+B^T

c)      (AB)^T = B^TA^T

d)      (rA)^T = rA^T

5)      Every nonzero m x n matrix is row equivalent to a unique matrix in reduced row echelon form.

6)      Let Ax = b and Cx =d be two linear systems each of m equations in n unknowns.  If the augmented matrices [A: b] and [C:d] of these systems are row equivalent, then both linear systems have exactly the same solutions. 

COROLLARY  If A and C are row equivalent m x n matrices, then the linear system Ax = 0 and Cx = 0  have exactly the same solutions.

7)      A homogeneous system of m equations in n unknowns always has a nontrivial solution if m<n, that is, if the number of unknowns exceeds the number of equations.

8)      If a matrix has an inverse, then the inverse is unique.

9)      Let A and B be nonsingular matrices

a)      A^-1 is nonsingular and (A^-1)^-1 = A

b)      AB is nonsingular and (AB)^-1 = B^-1A^-1

c)      (A^T)^-1 = (A^-1)^T

COROLLARY        If A₁, A₂. . . , A_r are n x n matrices, then A₁A₂. . . A_r is nonsingular and  (A₁A₂. . .A_r)^-1 = A_r^-1A_r-1^-1. . .A₁^-1

10)   Suppose that A and B are nxn matrices

a)      If AB = I_n, then BA = I_n

b)      If BA = I_n, then AB = I_n

11)  An n x n matrix is nonsingular if and only if is row equivalent to I_n

12)  if A is an n x n matrix, the homogeneous system Ax = 0 has a nontrivial solution if and       only if A is singular.

13)  If A is an n x n matrix, then A is nonsingular if and only if the linear system Ax = b has a unique solution for every n x 1 matrix b.

14)   det (A^T) = det (A)

15)  If matrix B results from matrix A by interchanging two rows (columns) of A, then det (B) = -det(A).

16)  If two rows (columns) of A are equal, then det(A)=0

17)  If a row (column) of A consists entirely of zeros, then det(A) = 0

18)  If B is obtained from A by multiplying a row (column) of A by a real number c, then det(B) = cdet(A).

19)  If B = [b_ij] is obtained from A = [a_ij] by adding to each element of the rth row (column) of A a constant c times the corresponding element of the sth row(column) r s of A, then det(B) = det(A).

20)  If a matrix A = [a_ij] is an upper (lower) triangular then det (A) = a₁₁a₂₂. . .a_nn  that is the determinant of a triangular matrix is the product of the elements on the main diagonal.

21)  det(AB) = det(A)det(B)
COROLLARY If A is nonsingular, then det (A) is not 0 and det (A^-1) = 1/det(A)

22)  Let A=[a_ij] be an n x n matrix. Then for each 1≤i≤n,
        det(A) = a_i1A_i1+a_i2A_i2+…+a_inA_in
(an expansion of det (A) about the ith row) and for each 1≤j≤n,
        det(A) = a_1jA_1j+a_1jA_1j. . .a_njA_nj
(expansion of det(A) about the ith row).

23)  If A = [a_ij] is a n x n matrix, then 
        a_i1A_k1+a_i2A_k2+. . . +a_inA_kn = 0  for i k
 and  
        a_1jA_1k+a_2jA_2k+. . . +a_njA_nk = 0 for j k

24)  If A=[a_ij] is an n x n matrix, then
        A(adj A) = (adj A)A =det(A) I_n.

25)  A matrix A is nonsingular if and only if det (A) is nonzero.
COROLLARY If A is an n x n matrix, then the homogeneous system Ax = 0 has a nontrivial solution if and only if det (A) = 0.

26) Cramer’s Rule Let

            a₁₁x₁+a₁₂x₂+. . . + a_1nx_n=b₁

            a₂₂x₁+a₂₂x₂+. . . + a_2nx_n=b₂

               :           :      :            :     :

               :           :      :            :     :

            a_n1x₁+a_n2x₂+. . . +a_nnx_n=b_n

      be a linear system of n equations in n unknowns and let A=[a_ij] be the coefficient matrix so that we can write the given system as Ax=b where

     b₁

     b₂

b= :

     :

     b_n

If det (A) is nonzero, then the system has the unique solution

x₁= det (A₁)/det(A), x₂=det(A₂)/det(A), . . . , x_n= det (A_n)/det(A),

where A_i is the matrix obtained from A by replacing the ith column of A by b.