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
A= 0 2
0 0

0
0 
4
7

4
B= 0
0

0 0
4 0 
0 4
Claim: For any m × n matrix A, AIn = A and Im A = A.

0
O= 0
0


0 0
0 0 
0 0
1 0
I3 =  0 1
0 0
k
k
Notation: We use A (k > 0) to denote the product of a square matrix A with itself k times.
A =
k
Y
i=1
q
Claim: For any p, q > 0, Ap Aq = Ap+q and (Ap ) = Apq .
Question: Does Ak B k = Ak B k in general?

0
0 
1
!
A
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.
• A matrix with real entries is symmetric if AT = A.
• A matrix with real entries is skew symmetric if AT = −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
A= 0 2
0 0

7
−5 
2
3
B. Partitioned Matrices:

4
B =  −3
1
1. Submatrices: Any matrix
original matrix A.

2
 −1
Example: Let A = 
 6
3

0
0
4
0 
−11 4


0 −2 4
C =  −2 0 5 
4
5 0

1
2
5
D =  −2
3 −10

−3
10 
0
formed by deleting some (but not all) of its rows and/or columns is a submatrix of the
0
4
−3
1

−5
7 

2 
0

2
If we delete the second column and the third row of A, we obtain the submatrix B =  −1
3

−5
7 
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.
Example: Let A =
1 4
3 7
and B =
− 75
3
5
4
5
− 51
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:
Since the inverse of an invertible matrix is unique, we will use A−1 to denote the inverse of a nonsingular matrix A.
1 2
2 3
.
and B =
Examples: Find the inverse of the following matrices, provided they exist: A =
2 4
4 5
Theorem 1.6 Suppose that A and B are nonsingular matrices. Then AB is nonsingular and (AB)−1 = B −1 A−1 .
Proof:
Theorem 1.7 If A is nonsingular, then its inverse A−1 is also nonsingular and A−1
Proof: Homework
Theorem 1.8 If A is nonsingular, then AT is also nonsingular and A−1
Proof:
T
= AT
−1
= A.
−1
D. Using the Inverse of a Matrix to Solve a Linear System of n equations in n unknowns:
Notice that if A~x = ~b and A is invertible, then A−1 (A~x) = A−1~b, or A−1 A ~x = A−1~b. Therefore, In ~x = A−1~b. Thus
~x = A−1~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.