Math 327
        Special Types of Matrices and Partitioned Matrices
        A. Definitions:
            • An n×n matrix A = [aij] is a diagonal matrix if aij = 0 whenever i 6= j.
            • An n×n matrix A = [aij] is a scalar matrix if it is a diagonal matrix whose diagonal elements are all equal.
            • The n×n identity matrix, denoted In is a scalar matrix whose diagonal entries are all equal to 1.
        Examples:
               −4 0 0                       4 0 0                       0 0 0                       1 0 0 
          A= 0 2 0                    B= 0 4 0                    O= 0 0 0                    I3 =  0   1  0 
                  0   0  4                     0  0 4                        0   0 0                        0 0 1
                         7
        Claim: For any m×n matrix A, AIn = A and ImA = A.
                                                                                                            k   !
                             k                                                                         k   Y
        Notation: We use A (k > 0) to denote the product of a square matrix A with itself k times.   A =      A
                                                                                                           i=1
                                   p q     p+q       p q     pq
        Claim: For any p;q > 0, A A = A        and (A ) = A .
                          k k
      k  k
        Question: Does A B       =A B ingeneral?
        Definitions:
            • An n×n matrix is upper triangular if aij = 0 whenever i > j.
            • An n×n matrix is lower triangular if aij = 0 whenever i < j.
                                                          T
            • A matrix with real entries is symmetric if A = A.
                                                                T
            • A matrix with real entries is skew symmetric if A = −A.
              Note: As a consequence of the definition, both symmetric and skew symmetric matrices must be square. In addition,
              the definition forces all diagonal entries of a skew symmetric matrix to be zero (since aii = −aii).
        Examples:
               −4 1      7                  4     0    0                0    −2 4                   1      2    −3 
          A= 0 2 −5                   B= −3       4    0          C= −2 0 5                   D= −2        5    10 
                  0   0   2                     1   −11 4                     4    5   0                    3    −10    0
                          3
        B. Partitioned Matrices:
           1. Submatrices: Any matrix formed by deleting some (but not all) of its rows and/or columns is a submatrix of the
              original matrix A.
                                  2      0   −5 
                                  −1     4    7 
              Example: Let A =                   
                                  6     −3    2 
                                     3    1    0
                                                                                                 2    −5 
              If we delete the second column and the third row of A, we obtain the submatrix B =  −1   7 
                                                                                                   3    0
           2. To form a partitioned matrix, we subdivide a given matrix A into submatrices by adding dotted lines between some
              portion of the rows and/or columns of the matrix.
        Example:
        Note: The augmented matrix representing a linear system is another example of a partitioned matrix.
         C. Singular and Nonsingular matrices:
         Definitions:
            • An n×n matrix A is nonsingular or invertible if there is an n×n matrix B such that AB = BA = In.
            • If no such matrix exists, we say that the matrix A is singular or noninvertible.
                               1 4               −7     4   
         Example: Let A =               and B =       5    5
                                3 7                   3   −1
                                                      5     5
         Claim: Singular matrices exist (a silly example is a zero matrix, but we will see other non-trivial examples).
         Theorem 1.5 The inverse of a matrix, if it exists, is unique.
         Proof:
                                                                           −1
         Since the inverse of an invertible matrix is unique, we will use A   to denote the inverse of a nonsingular matrix A.
         Examples: Find the inverse of the following matrices, provided they exist: A =  2       3  and B =  1     2 .
                                                                                               4 5                 2 4
                                                                                                               −1      −1 −1
         Theorem 1.6 Suppose that A and B are nonsingular matrices. Then AB is nonsingular and (AB)               =B A .
         Proof:
                                                                                               

                                                               −1                             −1 −1
         Theorem 1.7 If A is nonsingular, then its inverse A       is also nonsingular and A        =A.
         Proof: Homework
                                                                                   
        

                                                    T                             −1 T       T −1
         Theorem 1.8 If A is nonsingular, then A       is also nonsingular and A       = A
         Proof:
         D. Using the Inverse of a Matrix to Solve a Linear System of n equations in n unknowns:
                                                              −1           −1       −1 
          −1                        −1
                               ~                                             ~                       ~                         ~
         Notice that if A~x = b and A is invertible, then A     (A~x) = A    b, or  A A ~x = A b. Therefore, In~x = A           b. Thus
               −1
                 ~
         ~x = A   b.
         With this in mind, we can solve the original linear system by finding the inverse of the coefficient matrix A and then right
                             ~
         multiplying it with b to find the solution ~x.