MATH 311
Topics in Applied Mathematics I
Lecture 6:
Inverse matrix.
Identity matrix
Definition. The identity matrix (or unit matrix) is
a diagonal matrix with all diagonal entries equal to 1.
The n×n identity matrix is denoted In or simply I .


1 0 0
1 0
I1 = (1), I2 =
, I3 =  0 1 0  .
0 1
0 0 1


1 0 ... 0
0 1 . . . 0
In general, I =  .. .. . . . .. .
.
. .
0 0 ... 1
Theorem. Let A be an arbitrary m×n matrix.
Then Im A = AIn = A.
Inverse matrix
Let Mn (R) denote the set of all n×n matrices with
real entries. We can add, subtract, and multiply
elements of Mn (R). What about division?
Definition. Let A ∈ Mn (R). Suppose there exists
an n×n matrix B such that
AB = BA = In .
Then the matrix A is called invertible and B is
called the inverse of A (denoted A−1).
A non-invertible square matrix is called singular.
AA−1 = A−1A = I
Examples
1 1
1 −1
−1 0
A=
, B=
, C=
.
0 1
0 1
0 1
1 −1
1 0
AB =
=
,
0 1
0 1
1 −1
1 1
1 0
BA =
=
,
0 1
0 1
0 1
−1
0
−1
0
1
0
C2 =
=
.
0 1
0 1
0 1
1 1
0 1
Thus A−1 = B, B −1 = A, and C −1 = C .
Basic properties of inverse matrices
• If B = A−1 then A = B −1. In other words, if A
is invertible, so is A−1 , and A = (A−1)−1.
• The inverse matrix (if it exists) is unique.
Moreover, if AB = CA = I for some n×n matrices
B and C , then B = C = A−1.
Indeed, B = IB = (CA)B = C (AB) = CI = C .
• If n×n matrices A and B are invertible, so is
AB, and (AB)−1 = B −1A−1 .
(B −1 A−1 )(AB) = B −1 (A−1 A)B = B −1 IB = B −1 B = I ,
(AB)(B −1 A−1 ) = A(BB −1 )A−1 = AIA−1 = AA−1 = I .
−1 −1
• Similarly, (A1A2 . . . Ak )−1 = A−1
k . . . A2 A1 .
Inverting diagonal matrices
Theorem A diagonal matrix D = diag(d1, . . . , dn )
is invertible if and only if all diagonal entries are
nonzero: di 6= 0 for 1 ≤ i ≤ n.
If D is invertible then D −1 = diag(d1−1, . . . , dn−1).
−1 
d1−1 0
d1 0 . . . 0
 0 d −1
0 d ... 0
2



2
=

 ..
 .. .. . .
..
.
.
. .
 .
. .
.
0
0
0 0 . . . dn


... 0
... 0 

. . . ... 

−1
. . . dn
Inverting diagonal matrices
Theorem A diagonal matrix D = diag(d1, . . . , dn )
is invertible if and only if all diagonal entries are
nonzero: di 6= 0 for 1 ≤ i ≤ n.
If D is invertible then D −1 = diag(d1−1, . . . , dn−1).
Proof: If all di 6= 0 then, clearly,
diag(d1, . . . , dn ) diag(d1−1, . . . , dn−1) = diag(1, . . . , 1) = I ,
diag(d1−1, . . . , dn−1) diag(d1, . . . , dn ) = diag(1, . . . , 1) = I .
Now suppose that di = 0 for some i . Then for any
n×n matrix B the i th row of the matrix DB is a
zero row. Hence DB 6= I .
Inverting 2×2 matrices
Definition.
The determinant of a 2×2 matrix
a b
A=
is det A = ad − bc.
c d
a b
Theorem A matrix A =
is invertible if
c d
and only if det A 6= 0.
If det A 6= 0 then
−1
1
a b
d −b
=
.
c d
a
ad − bc −c
a b
Theorem A matrix A =
is invertible if
c d
and only if det A 6= 0. If det A 6= 0 then
−1
1
a b
d −b
=
.
c d
a
ad − bc −c
d −b
Proof: Let B =
. Then
−c
a
ad −bc
0
AB = BA =
= (ad − bc)I2 .
0
ad −bc
In the case det A 6= 0, we have A−1 = (det A)−1 B.
In the case det A = 0, the matrix A is not invertible as
otherwise AB = O =⇒ A−1 (AB) = A−1 O = O
=⇒ (A−1 A)B = O =⇒ I2 B = O =⇒ B = O
=⇒ A = O, but the zero matrix is singular.
Problem. Solve a system
4x + 3y = 5,
3x + 2y = −1.
This system is equivalent to a matrix equation Ax = b,
4 3
x
5
where A =
, x=
, b=
.
3 2
y
−1
We have det A = − 1 6= 0. Hence A is invertible.
Ax = b =⇒ A−1 (Ax) = A−1 b =⇒ (A−1 A)x = A−1 b
=⇒ x = A−1 b.
Conversely, x = A−1 b =⇒ Ax = A(A−1 b) = (AA−1 )b = b.
−1 1
x
4 3
5
2 −3
5
−13
=
=
=
y
3 2
−1
4
−1
19
−1 −3
System of n linear equations in n variables:

a11x1 + a12x2 + · · · + a1n xn = b1



a21x1 + a22x2 + · · · + a2n xn = b2
⇐⇒ Ax = b,
·
·
·
·
·
·
·
·
·



an1 x1 + an2 x2 + · · · + ann xn = bn
where

a11 a12
a a
 21 22
A =  ..
..
 .
.
an1 an2

. . . a1n
. . . a2n 

,
. . . ... 

. . . ann
 
 
x1
b1
x 
b 
 2
 2
x =  .. , b =  ..  .
.
.
xn
bn
Theorem If the matrix A is invertible then the
system has a unique solution, which is x = A−1b.
General results on inverse matrices
Theorem 1 Given an n×n matrix A, the following
conditions are equivalent:
(i) A is invertible;
(ii) x = 0 is the only solution of the matrix equation Ax = 0;
(iii) the matrix equation Ax = b has a unique solution for
any n-dimensional column vector b;
(iv) the row echelon form of A has no zero rows;
(v) the reduced row echelon form of A is the identity matrix.
Theorem 2 Suppose that a sequence of elementary row
operations converts a matrix A into the identity matrix.
Then the same sequence of operations converts the identity
matrix into the inverse matrix A−1 .
Row echelon form of a square matrix:















∗

∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗

∗ 


∗ 


∗ 

∗ 


∗ 

∗
invertible case















∗

∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗

∗ 


∗ 


∗ 

∗ 




∗
noninvertible case