Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
MATH 4A - Linear Algebra with Applications
Lecture 9: Matrix inverses
Substitute lecturer: Professor Priyam Patel
19 April 2019
Reading: §2.2, 2.3, 3.1, 3.2 from Lay, 5th ed.
Recommended problems from §2.2: 1, 3, 5, 7, 9, 10, 13,
Recommended problems from §2.3: 1, 3, 5, 7, 11, 12, 13, 15, 17,
21
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Lecture plan
1 Inverse of a matrix
2 An algorithm for finding matrix inverses
3 Other characterizations of invertible matrices
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Motivation
When we do usual algebra in a single variable, we can “cancel” or
“divide out.” For instance, to solve the equation
2x = 4
we can divide both sides by 2—or, in other words, we can multiply
both sides by the multiplicative inverse of 2:
(2−1 )2x = (2−1 )4
to see that
x = 2.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Motivation
In the previous lecture, we developed algebraic rules for working
with matrices. One important caveat about the algebra of matrices
was that there was no cancellation law: there exist matrices
A, B, C such that
AB = AC but B 6= C .
The fundamental issue is that a matrix A does not always have an
inverse with respect to the operation of matrix multiplication.
However, some matrices do have inverses. And it turns out, these
matrices are very useful. (And, in fact, we’ve already been using
them implicitly any time we did a row operation.)
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Definition
An n × n square matrix A is called invertible if there exists a n × n
square matrix C such that
AC = CA = I = In ,
where I = In is the n × n identity matrix. We call such a C an
inverse of A.
In fact, if an inverse C of A exists, then C is unique. This is
because if B is another inverse, then
B = BI = B(AC ) = (BA)C = IC = C .
If A is invertible, we will denote its unique inverse by A−1 , so that
AA−1 = A−1 A = I .
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Basic questions:
Given a matrix A, how can we decide if A is invertible? Once we
know A is invertible, how do we compute its inverse A−1 ?
We’ll address these questions today.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Example
If
1 1
A=
0 1
we can check that the inverse is
−1 1 −1
A = .
0 1
−1 1 1 1 −1 1 · 1 + 1 · 0 1 · (−1) + 1 · 1
AA = =
0 1 0 1 0 · 1 + 1 · 0 0 · (−1) + 1 · 1
1 0
= =I
0 1
−1 1 −1 1 1 1 · 1 + (−1) · 0 1 · 1 + (−1) · (1)
A A= =
0 1 0 1 0·1+1·1 0·1+1·1
1 0
= =I
0 1
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Nonexample
No matter how hard we try, we can never find an inverse for the
matrix
1 1
B= .
0 0
Why not? Let
a b
C=
c d
be some matrix. Then
1 1 a b a b
BC = = .
0 0 c d 0 0
So no matter what we pick for a, b, c, d (that is, no matter
which
a b
matrix C we choose) we can never get the matrix BC =
0 0
1 0
to equal the identity I2 = . Thus B is not invertible.
0 1
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Understanding invertibility and inverses of 2 × 2 matrices is
easy
Theorem
a b
Let A = . Then A is invertible if and only if ad − bc 6= 0.
c d
If A is invertible, then its inverse is
1 d −b
A−1 = .
ad − bc −c a
We call the number ad − bc the determinant of A, and write
det A = ad − bc.
Thus, a 2 × 2 matrix A is invertible if and only if det A 6= 0.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Let’s prove one half of the theorem
a b
Let A = be a 2 × 2 matrix, and suppose det A 6= 0. Then
c d
it makes sense to divide by det A = ad − bc, so the matrix
1 d −b
ad − bc −c a
is well defined. To see that this matrix is indeed the inverse of A,
we just compute:
1 d −b 1 d −b
A = A
ad − bc −c a ad − bc −c a
1 a b d −b
=
ad − bc c d −c a
1 ad − bc 0 1 0
= = .
ad − bc 0 ad − bc 0 1
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
A similar calculation shows
1 d −b 1 0
A= .
ad − bc −c a 0 1
We conclude that
−1 1 d −b
A = .
ad − bc −c a
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
One reason we like invertibility: Linear systems with
invertible coefficient matrix are especially easy to solve
Theorem
If A is an invertible n × n matrix, then for each b in Rn , the
equation Ax = b has the unique solution x = A−1 b.
Proof: given A and b, it’s easy to see that x = A−1 b is a solution:
Ax = AA−1 b = In b = b.
To see that A−1 b is the unique solution, let u be another solution.
Then
Au = b = AA−1 b,
and we can “cancel” the A on the left by multiplying each side of
the equation by A−1 on the left:
A−1 Au = A−1 AA−1 b
u = A−1 b.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
iClicker 1
Using matrix inverses, find the first entry of the solution to the
equation
1 1 x1 0
= .
0 1 x2 2
Recall from an earlier slide that
−1
1 1 1 −1
=
0 1 0 1
(a) 0
(b) 1
(c) -1
(d) -2
(e) 2
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Useful algebraic identities about matrix inverses
Theorem
Let A and B be two invertible n × n matrices. Then
(a) A−1 is also invertible, and (A−1 )−1 = A
(b) (AB)−1 = B −1 A−1
(c) (AT )−1 = (A−1 )T
When using identity (b), it’s especially important to remember
that matrix multiplication is not commutative.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
iClicker 2
What is the (2, 2)-entry of the matrix (AB)−1 , where
−1 4 −1 −1 0 1
A = B = ?
0 1 2 3
(a) -2
(b) 1
(c) 2
(d) 3
(e) 8
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Elementary matrices
An elementary matrix is one that is obtained by performing a
single elementary row operation on an identity matrix.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Examples of elementary matrices
1 0 0 0 1 0 1 0 0
E1 = 0 1 0 E 2 = 1 0 0 E 3 = 0 1 0
−4 0 1 0 0 1 0 0 5
E1 is the result of replacing R3 with R3 − 4R1 in I3 . E2 is the result
of swapping rows R1 and R2 in I3 . And E3 is the result of scaling
R3 by the nonzero scalar 5.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Left multiplying a matrix A by an elementary matrix E is
the same thing as performing an elementary row operation
on A
Example: let
a b c
A = d e f .
g h i
Then
1 0 0 a b c a b c
E1 A = 0 1 0 d e f = d e f
−4 0 1 g h i g − 4a h − 4b i − 4c
is the same thing we would get if we replaced R3 of A with
R3 − 4R1 .
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Two important observations
1 If an elementary row operation is performed on an m × n
matrix A, the resulting matrix can be writtten as EA, where
the m × m matrix E is created by performing the same row
operation on Im .
2 Each elementary matrix E is invertible. The inverse of E is
the elementary matrix of the same type that transforms E
back into I .
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
iClicker 3
Which of the following is the inverse of the elementary matrix
1 0 0
E1 = 0 1 0?
−4 0 1
−4 0 1
(a) 0 1 0
1 0 0
1 0 0
(b) 0 1 0
4 0 1
4 0 1
(c) 0 1 0
1 0 0
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Most important slide of the day
Theorem
An n × n matrix A is invertible if and only if A is row equivalent to
the identity matrix In . In this case, any sequence of elementary row
operations that reduces A to In also transforms In into A−1 .
We won’t prove this theorem, but we will use it many times. In
particular, it implies that there is an algorithm for finding matrix
inverses.
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Algorithm for finding matrix inverses
1 If A is an n × n matrix, form the “block-augmented” matrix
A In .
2 Perform row operations
on A In , with the goal to get it in
the form In C .
3 If at any point you get a row where the first n entries are 0,
but one of the last n entries is nonzero, then A is not
invertible, so you can stop looking for the inverse.
4 Otherwise, following the usual row reduction
algorithm for A
but applying the operations to A In will result in a matrix
of the form In C . In this case, C = A−1 .
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
Example
1 2
Let’s find the inverse of A = using the algorithm.
3 4
1 2 1 0 R2 7→R2 −3R2 1 2 1 0
−
−−−−−−− →
3 4 0 1 0 −2 −3 1
1 0 −2 1 R2 7→ −1
R1 7→R1 +R2 2
R2 1 0 −2 1
−−−−−−−→ −−−−−−→ −1
0 −2 −3 1 0 1 23 2
Thus
−1 −2 1
A = 3 .
2 − 21
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
There are many ways to characterize invertible matrices
Theorem
Let A be a n × n square matrix. Then the following are all
equivalent
(a) A is invertible
(b) A is row equivalent to In
(c) A has n pivot positions
(d) Ax = 0 has only the trivial solution
(e) the columns of A are linearly independent
(f) The linear transformation A : Rn → Rn is one-to-one
(g) the equation Ax = b is consistent for all b in Rn
(h) the columns of A span Rn
(i) The linear transformation A : Rn → Rn is onto
(j) AT is invertible
Inverse of a matrix An algorithm for finding matrix inverses Other characterizations of invertible matrices
iClicker 4
Is the matrix
1 0 2
A = 0 1 3
1 1 5
invertible? Why?
(a) Yes, because none of the rows are all 0
(b) Yes, because none of the columns are all 0
(c) Yes, because the columns are linearly independent
(d) No, because the columns are linearly dependent
(e) No, because the equation Ax = 0 is inconsistent.
