From A First Course in Linear Algebra
Version 2.00
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
We begin with a familiar example, performed in a novel way.
Example SABMI
Solutions to Archetype B with a matrix inverse
Archetype B is the system of m = 3
linear equations in n = 3
variables,
By Theorem SLEMM we can represent this system of equations as
Ax = b

where
We’ll pull a rabbit out of our hat and present the 3 × 3 matrix B,
B = \left [\array{
−10&−12&−9
\cr
{13\over
2} & 8 & {11\over
2}
\cr
{5\over
2} & 3 & {5\over
2} } \right ]

and note that
BA = \left [\array{
−10&−12&−9
\cr
{13\over
2} & 8 & {11\over
2}
\cr
{5\over
2} & 3 & {5\over
2} } \right ]\left [\array{
−7&−6&−12
\cr
5 & 5 & 7
\cr
1 & 0 & 4 } \right ] = \left [\array{
1&0&0
\cr
0&1&0
\cr
0&0&1} \right ]

Now apply this computation to the problem of solving the system of equations,
So we have
x = Bb = \left [\array{
−10&−12&−9
\cr
{13\over
2} & 8 & {11\over
2}
\cr
{5\over
2} & 3 & {5\over
2} } \right ]\left [\array{
−33
\cr
24
\cr
5 } \right ] = \left [\array{
−3
\cr
5
\cr
2 } \right ]

So with the help and assistance of B we have been able to determine a solution to the system represented by Ax = b through judicious use of matrix multiplication. We know by Theorem NMUS that since the coefficient matrix in this example is nonsingular, there would be a unique solution, no matter what the choice of b. The derivation above amplifies this result, since we were forced to conclude that x = Bb and the solution couldn’t be anything else. You should notice that this argument would hold for any particular value of b. ⊠
The matrix B of the previous example is called the inverse of A. When A and B are combined via matrix multiplication, the result is the identity matrix, which can be inserted “in front” of x as the first step in finding the solution. This is entirely analogous to how we might solve a single linear equation like 3x = 12.
x = 1x = \left ({1\over
3}\left (3\right )\right )x = {1\over
3}\left (3x\right ) = {1\over
3}\left (12\right ) = 4

Here we have obtained a solution by employing the “multiplicative inverse” of 3, {3}^{−1} = {1\over 3}. This works fine for any scalar multiple of x, except for zero, since zero does not have a multiplicative inverse. For matrices, it is more complicated. Some matrices have inverses, some do not. And when a matrix does have an inverse, just how would we compute it? In other words, just where did that matrix B in the last example come from? Are there other matrices that might have worked just as well?
Definition MI
Matrix Inverse
Suppose A and
B are square
matrices of size n
such that AB = {I}_{n}
and BA = {I}_{n}. Then
A is invertible and
B is the inverse of
A. In this situation,
we write B = {A}^{−1}.
(This definition contains Notation MI.) △
Notice that if B is the inverse of A, then we can just as easily say A is the inverse of B, or A and B are inverses of each other.
Not every square matrix has an inverse. In Example SABMI the matrix B is the inverse the coefficient matrix of Archetype B. To see this it only remains to check that AB = {I}_{3}. What about Archetype A? It is an example of a square matrix without an inverse.
Example MWIAA
A matrix without an inverse, Archetype A
Consider the coefficient matrix from Archetype A,
A = \left [\array{
1&−1&2
\cr
2& 1 &1
\cr
1& 1 &0 } \right ]

Suppose that A is invertible and does have an inverse, say B. Choose the vector of constants
b = \left [\array{
1
\cr
3
\cr
2 } \right ]

and consider the system of equations ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right ). Just as in Example SABMI, this vector equation would have the unique solution x = Bb.
However, the system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right ) is inconsistent. Form the augmented matrix \left [\left .A\kern 1.95872pt \right \vert \kern 1.95872pt b\right ] and rowreduce to
\left [\array{
\text{1}&0& 1 &0
\cr
0&\text{1}&−1&0
\cr
0&0& 0 &\text{1} } \right ]

which allows to recognize the inconsistency by Theorem RCLS.
So the assumption of A’s inverse leads to a logical inconsistency (the system can’t be both consistent and inconsistent), so our assumption is false. A is not invertible.
Its possible this example is less than satisfying. Just where did that particular choice of the vector b come from anyway? Stay tuned for an application of the future Theorem CSCS in Example CSAA. ⊠
Let’s look at one more matrix inverse before we embark on a more systematic study.
Example MI
Matrix inverse
Consider the matrices,
Then
so by Definition MI, we can say that A is invertible and write B = {A}^{−1}. ⊠
We will now concern ourselves less with whether or not an inverse of a matrix exists, but instead with how you can find one when it does exist. In Section MINM we will have some theorems that allow us to more quickly and easily determine just when a matrix is invertible.
We’ve seen that the matrices from Archetype B and Archetype K both have inverses, but these inverse matrices have just dropped from the sky. How would we compute an inverse? And just when is a matrix invertible, and when is it not? Writing a putative inverse with {n}^{2} unknowns and solving the resultant {n}^{2} equations is one approach. Applying this approach to 2 × 2 matrices can get us somewhere, so just for fun, let’s do it.
Theorem TTMI
TwobyTwo Matrix Inverse
Suppose
A = \left [\array{
a&b
\cr
c&d } \right ]

Then A is invertible if and only if ad − bc\mathrel{≠}0. When A is invertible, then
{
A}^{−1} = {1\over
ad − bc}\left [\array{
d &−b
\cr
−c& a } \right ]

Proof ( ⇐) Assume that ad − bc\mathrel{≠}0. We will use the definition of the inverse of a matrix to establish that A has inverse (Definition MI). Note that if ad − bc\mathrel{≠}0 then the displayed formula for {A}^{−1} is legitimate since we are not dividing by zero). Using this proposed formula for the inverse of A, we compute
By Definition MI this is sufficient to establish that A is invertible, and that the expression for {A}^{−1} is correct.
( ⇒) Assume that A is invertible, and proceed with a proof by contradiction (Technique CD), by assuming also that ad − bc = 0. This translates to ad = bc. Let
B = \left [\array{
e&f
\cr
g&h } \right ]

be a putative inverse of A. This means that
{
I}_{2} = AB = \left [\array{
a&b
\cr
c&d } \right ]\left [\array{
e&f
\cr
g&h } \right ] = \left [\array{
ae + bg&af + bh
\cr
ce + dg&cf + dh } \right ]

Working on the matrices on both ends of this equation, we will multiply the top row by c and the bottom row by a.
\left [\array{
c&0
\cr
0&a } \right ] = \left [\array{
ace + bcg&acf + bch
\cr
ace + adg&acf + adh } \right ]

We are assuming that ad = bc, so we can replace two occurences of ad by bc in the bottom row of the right matrix.
\left [\array{
c&0
\cr
0&a } \right ] = \left [\array{
ace + bcg&acf + bch
\cr
ace + bcg&acf + bch } \right ]

The matrix on the right now has two rows that are identical, and therefore the same must be true of the matrix on the left. Given the form of the matrix on the left, identical rows implies that a = 0 and c = 0.
With this information, the product AB becomes
\left [\array{
1&0
\cr
0&1 } \right ] = {I}_{2} = AB = \left [\array{
ae + bg&af + bh
\cr
ce + dg&cf + dh } \right ] = \left [\array{
bg&bh
\cr
dg&dh } \right ]

So bg = dh = 1 and thus b,g,d,h are all nonzero. But then bh and dg (the “other corners”) must also be nonzero, so this is (finally) a contradiction. So our assumption was false and we see that ad − bc\mathrel{≠}0 whenever A has an inverse. ■
There are several ways one could try to prove this theorem, but there is a continual temptation to divide by one of the eight entries involved (a through f), but we can never be sure if these numbers are zero or not. This could lead to an analysis by cases, which is messy, messy, messy. Note how the above proof never divides, but always multiplies, and how zero/nonzero considerations are handled. Pay attention to the expression ad − bc, as we will see it again in a while (Chapter D).
This theorem is cute, and it is nice to have a formula for the inverse, and a condition that tells us when we can use it. However, this approach becomes impractical for larger matrices, even though it is possible to demonstrate that, in theory, there is a general formula. (Think for a minute about extending this result to just 3 × 3 matrices. For starters, we need 18 letters!) Instead, we will work columnbycolumn. Let’s first work an example that will motivate the main theorem and remove some of the previous mystery.
Example CMI
Computing a matrix inverse
Consider the matrix defined in Example MI as,
A = \left [\array{
1 & 2 & 1 & 2 & 1
\cr
−2&−3& 0 &−5&−1
\cr
1 & 1 & 0 & 2 & 1
\cr
−2&−3&−1&−3&−2
\cr
−1&−3&−1&−3& 1 } \right ]

For its inverse, we desire a matrix B so that AB = {I}_{5}. Emphasizing the structure of the columns and employing the definition of matrix multiplication Definition MM,
Equating the matrices columnbycolumn we have
Since the matrix B is what we are trying to compute, we can view each column, {B}_{i}, as a column vector of unknowns. Then we have five systems of equations to solve, each with 5 equations in 5 variables. Notice that all 5 of these systems have the same coefficient matrix. We’ll now solve each system in turn,
We can now collect our 5 solution vectors into the matrix B,
By this method, we know that AB = {I}_{5}. Check that BA = {I}_{5}, and then we will know that we have the inverse of A. ⊠
Notice how the five systems of equations in the preceding example were all solved by exactly the same sequence of row operations. Wouldn’t it be nice to avoid this obvious duplication of effort? Our main theorem for this section follows, and it mimics this previous example, while also avoiding all the overhead.
Theorem CINM
Computing the Inverse of a Nonsingular Matrix
Suppose A is a nonsingular
square matrix of size n.
Create the n × 2n matrix
M by placing the
n × n identity matrix
{I}_{n} to the right of
the matrix A. Let
N be a matrix that is
rowequivalent to M
and in reduced rowechelon form. Finally, let
J be the matrix
formed from the final n
columns of N.
Then AJ = {I}_{n}.
□
Proof A is nonsingular, so by Theorem NMRRI there is a sequence of row operations that will convert A into {I}_{n}. It is this same sequence of row operations that will convert M into N, since having the identity matrix in the first n columns of N is sufficient to guarantee that N is in reduced rowechelon form.
If we consider the systems of linear equations, ℒS\kern 1.95872pt \left (A,\kern 1.95872pt {e}_{i}\right ), 1 ≤ i ≤ n, we see that the aforementioned sequence of row operations will also bring the augmented matrix of each of these systems into reduced rowechelon form. Furthermore, the unique solution to ℒS\kern 1.95872pt \left (A,\kern 1.95872pt {e}_{i}\right ) appears in column n + 1 of the rowreduced augmented matrix of the system and is identical to column n + i of N. Let {N}_{1},\kern 1.95872pt {N}_{2},\kern 1.95872pt {N}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {N}_{2n} denote the columns of N. So we find,
as desired. ■
We have to be just a bit careful here about both what this theorem says and what it doesn’t say. If A is a nonsingular matrix, then we are guaranteed a matrix B such that AB = {I}_{n}, and the proof gives us a process for constructing B. However, the definition of the inverse of a matrix (Definition MI) requires that BA = {I}_{n} also. So at this juncture we must compute the matrix product in the “opposite” order before we claim B as the inverse of A. However, we’ll soon see that this is always the case, in Theorem OSIS, so the title of this theorem is not inaccurate.
What if A is singular? At this point we only know that Theorem CINM cannot be applied. The question of A’s inverse is still open. (But see Theorem NI in the next section.) We’ll finish by computing the inverse for the coefficient matrix of Archetype B, the one we just pulled from a hat in Example SABMI. There are more examples in the Archetypes (Appendix A) to practice with, though notice that it is silly to ask for the inverse of a rectangular matrix (the sizes aren’t right) and not every square matrix has an inverse (remember Example MWIAA?).
Example CMIAB
Computing a matrix inverse, Archetype B
Archetype B has a coefficient matrix given as
once we check that {B}^{−1}B = {I}_{ 3} (the product in the opposite order is a consequence of the theorem). ⊠
While we can use a rowreducing procedure to compute any needed inverse, most computational devices have a builtin procedure to compute the inverse of a matrix straightaway. See: Computation MI.MMA Computation MI.SAGE.
The inverse of a matrix enjoys some nice properties. We collect a few here. First, a matrix can have but one inverse.
Theorem MIU
Matrix Inverse is Unique
Suppose the square matrix A
has an inverse. Then {A}^{−1}
is unique. □
Proof As described in Technique U, we will assume that A has two inverses. The hypothesis tells there is at least one. Suppose then that B and C are both inverses for A. Then, repeated use of Definition MI and Theorem MMIM plus one application of Theorem MMA gives
So we conclude that B and C are the same, and cannot be different. So any matrix that acts like an inverse, must be the inverse. ■
When most of us dress in the morning, we put on our socks first, followed by our shoes. In the evening we must then first remove our shoes, followed by our socks. Try to connect the conclusion of the following theorem with this everyday example.
Theorem SS
Socks and Shoes
Suppose A and
B are invertible
matrices of size n.
Then {(AB)}^{−1} = {B}^{−1}{A}^{−1} and
AB is an invertible
matrix. □
Proof At the risk of carrying our everyday analogies too far, the proof of this theorem is quite easy when we compare it to the workings of a dating service. We have a statement about the inverse of the matrix AB, which for all we know right now might not even exist. Suppose AB was to sign up for a dating service with two requirements for a compatible date. Upon multiplication on the left, and on the right, the result should be the identity matrix. In other words, AB’s ideal date would be its inverse.
Now along comes the matrix {B}^{−1}{A}^{−1} (which we know exists because our hypothesis says both A and B are invertible and we can form the product of these two matrices), also looking for a date. Let’s see if {B}^{−1}{A}^{−1} is a good match for AB. First they meet at a noncommittal neutral location, say a coffee shop, for quiet conversation:
}
So the matrix {B}^{−1}{A}^{−1} has met all of the requirements to be AB’s inverse (date) and with the ensuing marriage proposal we can announce that {(AB)}^{−1} = {B}^{−1}{A}^{−1}. ■
Theorem MIMI
Matrix Inverse of a Matrix Inverse
Suppose A is an
invertible matrix. Then {A}^{−1}
is invertible and {({A}^{−1})}^{−1} = A.
□
Proof As with the proof of Theorem SS, we examine if A is a suitable inverse for {A}^{−1} (by definition, the opposite is true).
}
The matrix A has met all the requirements to be the inverse of {A}^{−1}, and so is invertible and we can write A = {({A}^{−1})}^{−1}. ■
Theorem MIT
Matrix Inverse of a Transpose
Suppose A is an
invertible matrix. Then {A}^{t}
is invertible and {({A}^{t})}^{−1} = {({A}^{−1})}^{t}.
□
Proof As with the proof of Theorem SS, we see if {({A}^{−1})}^{t} is a suitable inverse for {A}^{t}. Apply Theorem MMT to see that
}
The matrix {({A}^{−1})}^{t} has met all the requirements to be the inverse of {A}^{t}, and so is invertible and we can write {({A}^{t})}^{−1} = {({A}^{−1})}^{t}. ■
Theorem MISM
Matrix Inverse of a Scalar Multiple
Suppose A is an invertible
matrix and α is a
nonzero scalar. Then {\left (αA\right )}^{−1} = {1\over
α}{A}^{−1}
and αA is
invertible. □
Proof As with the proof of Theorem SS, we see if {1\over α}{A}^{−1} is a suitable inverse for αA.
}
The matrix {1\over α}{A}^{−1} has met all the requirements to be the inverse of αA, so we can write {\left (αA\right )}^{−1} = {1\over α}{A}^{−1}. ■
Notice that there are some likely theorems that are missing here. For example, it would be tempting to think that {(A + B)}^{−1} = {A}^{−1} + {B}^{−1}, but this is false. Can you find a counterexample? (See Exercise MISLE.T10.)
\left [\array{
4&10
\cr
2& 6
} \right ]

\left [\array{
2 & 3 & 1
\cr
1 &−2&−3
\cr
−2& 4 & 6
} \right ]

C21 Verify that B is the inverse of A.
Contributed by Robert Beezer Solution [644]
C22 Recycle the matrices A and B from Exercise MISLE.C21 and set
Employ the matrix B to solve
the two linear systems ℒS\kern 1.95872pt \left (A,\kern 1.95872pt c\right )
and ℒS\kern 1.95872pt \left (A,\kern 1.95872pt d\right ).
Contributed by Robert Beezer Solution [644]
C23 If it exists, find the inverse of the 2 × 2 matrix
and check your answer. (See Theorem TTMI.)
Contributed by Robert Beezer
C24 If it exists, find the inverse of the 2 × 2 matrix
and check your answer. (See Theorem TTMI.)
Contributed by Robert Beezer
C25 At the conclusion of Example CMI, verify that
BA = {I}_{5} by
computing the matrix product.
Contributed by Robert Beezer
C26 Let
D = \left [\array{
1 &−1& 3 &−2&1
\cr
−2& 3 &−5& 3 &0
\cr
1 &−1& 4 &−2&2
\cr
−1& 4 &−1& 0 &4
\cr
1 & 0 & 5 &−2&5 } \right ]

Compute the inverse of D,
{D}^{−1}, by forming
the 5 × 10
matrix \left [\left .D\kern 1.95872pt \right \vert \kern 1.95872pt {I}_{5}\right ]
and rowreducing (Theorem CINM). Then use a calculator to compute
{D}^{−1}
directly.
Contributed by Robert Beezer Solution [645]
C27 Let
E = \left [\array{
1 &−1& 3 &−2& 1
\cr
−2& 3 &−5& 3 &−1
\cr
1 &−1& 4 &−2& 2
\cr
−1& 4 &−1& 0 & 2
\cr
1 & 0 & 5 &−2& 4 } \right ]

Compute the inverse of E,
{E}^{−1}, by forming
the 5 × 10
matrix \left [\left .E\kern 1.95872pt \right \vert \kern 1.95872pt {I}_{5}\right ]
and rowreducing (Theorem CINM). Then use a calculator to compute
{E}^{−1}
directly.
Contributed by Robert Beezer Solution [645]
C28 Let
C = \left [\array{
1 & 1 & 3 & 1
\cr
−2&−1&−4&−1
\cr
1 & 4 &10& 2
\cr
−2& 0 &−4& 5 } \right ]

Compute the inverse of C,
{C}^{−1}, by forming
the 4 × 8
matrix \left [\left .C\kern 1.95872pt \right \vert \kern 1.95872pt {I}_{4}\right ]
and rowreducing (Theorem CINM). Then use a calculator to compute
{C}^{−1}
directly.
Contributed by Robert Beezer Solution [645]
C40 Find all solutions to the system of equations below, making use of the matrix inverse found in Exercise MISLE.C28.
Contributed by Robert Beezer Solution [646]
C41 Use the inverse of a matrix to find all the solutions to the following system of equations.
Contributed by Robert Beezer Solution [647]
C42 Use a matrix inverse to solve the linear system of equations.
Contributed by Robert Beezer Solution [648]
T10 Construct an example to demonstrate that
{(A + B)}^{−1} = {A}^{−1} + {B}^{−1} is not true for all
square matrices A
and B
of the same size.
Contributed by Robert Beezer Solution [649]
C21 Contributed by Robert Beezer Statement [637]
Check that both matrix products (Definition MM)
AB and
BA equal the
4 × 4 identity
matrix {I}_{4}
(Definition IM).
C22 Contributed by Robert Beezer Statement [637]
Represent each of the two systems by a vector equality,
Ax = c and
Ay = d.
Then in the spirit of Example SABMI, solutions are given by
Notice how we could solve many more systems having A as the coefficient matrix, and how each such system has a unique solution. You might check your work by substituting the solutions back into the systems of equations, or forming the linear combinations of the columns of A suggested by Theorem SLSLC.
C26 Contributed by Robert Beezer Statement [639]
The inverse of D
is
{
D}^{−1} = \left [\array{
−7&−6&−3& 2 & 1
\cr
−7&−4& 2 & 2 &−1
\cr
−5&−2& 3 & 1 &−1
\cr
−6&−3& 1 & 1 & 0
\cr
4 & 2 &−2&−1& 1 } \right ]

C27 Contributed by Robert Beezer Statement [640]
The matrix E
has no inverse, though we do not yet have a theorem that allows us
to reach this conclusion. However, when rowreducing the matrix
\left [\left .E\kern 1.95872pt \right \vert \kern 1.95872pt {I}_{5}\right ], the first 5 columns will
not rowreduce to the 5 × 5
identity matrix, so we are a t a loss on how we might compute
the inverse. When requesting that your calculator compute
{E}^{−1}, it should give some
indication that E
does not have an inverse.
C28 Contributed by Robert Beezer Statement [640]
Employ Theorem CINM,
\left [\array{
1 & 1 & 3 & 1 &1&0&0&0
\cr
−2&−1&−4&−1&0&1&0&0
\cr
1 & 4 &10& 2 &0&0&1&0
\cr
−2& 0 &−4& 5 &0&0&0&1 } \right ]\mathop{\longrightarrow}\limits_{}^{\text{RREF}}\left [\array{
\text{1}&0&0&0& 38 & 18 & −5 &−2
\cr
0&\text{1}&0&0& 96 & 47 &−12&−5
\cr
0&0&\text{1}&0&−39&−19& 5 & 2
\cr
0&0&0&\text{1}&−16& −8 & 2 & 1 } \right ]

And therefore we see that C is nonsingular (C rowreduces to the identity matrix, Theorem NMRRI) and by Theorem CINM,
{
C}^{−1} = \left [\array{
38 & 18 & −5 &−2
\cr
96 & 47 &−12&−5
\cr
−39&−19& 5 & 2
\cr
−16& −8 & 2 & 1 } \right ]

C40 Contributed by Robert Beezer Statement [641]
View this system as ℒS\kern 1.95872pt \left (C,\kern 1.95872pt b\right ),
where C is the
4 × 4 matrix from
Exercise MISLE.C28 and b = \left [\array{
−4
\cr
4
\cr
−20
\cr
9 } \right ].
Since C
was seen to be nonsingular in Exercise MISLE.C28 Theorem SNCM says the
solution, which is unique by Theorem NMUS, is given by
{
C}^{−1}b = \left [\array{
38 & 18 & −5 &−2
\cr
96 & 47 &−12&−5
\cr
−39&−19& 5 & 2
\cr
−16& −8 & 2 & 1 } \right ]\left [\array{
−4
\cr
4
\cr
−20
\cr
9 } \right ] = \left [\array{
2
\cr
−1
\cr
−2
\cr
1 } \right ]

Notice that this solution can be easily checked in the original system of equations.
C41 Contributed by Robert Beezer Statement [642]
The coefficient matrix of this system of equations is
A = \left [\array{
1 & 2 &−1
\cr
2 & 5 &−1
\cr
−1&−4& 0 } \right ]

and the vector of constants is b = \left [\array{ −3 \cr −4 \cr 2 } \right ]. So by Theorem SLEMM we can convert the system to the form Ax = b. Rowreducing this matrix yields the identity matrix so by Theorem NMRRI we know A is nonsingular. This allows us to apply Theorem SNCM to find the unique solution as
x = {A}^{−1}b = \left [\array{
−4& 4 & 3
\cr
1 &−1&−1
\cr
−3& 2 & 1 } \right ]\left [\array{
−3
\cr
−4
\cr
2 } \right ] = \left [\array{
2
\cr
−1
\cr
3 } \right ]

Remember, you can check this solution easily by evaluating the matrixvector product Ax (Definition MVP).
C42 Contributed by Robert Beezer Statement [642]
We can reformulate the linear system as a vector equality with a matrixvector
product via Theorem SLEMM. The system is then represented by
Ax = b
where
According to Theorem SNCM, if A is nonsingular then the (unique) solution will be given by {A}^{−1}b. We attempt the computation of {A}^{−1} through Theorem CINM, or with our favorite computational device and obtain,
So by Theorem NI, we know A is nonsingular, and so the unique solution is
T10 Contributed by Robert Beezer Statement [643]
Let D be
any 2 × 2
matrix that has an inverse (Theorem TTMI can help you construct such a matrix,
{I}_{2} is a simple
choice). Set A = D
and B = (−1)D.
While {A}^{−1} and
{B}^{−1} both exist,
what is {\left (A + B\right )}^{−1}?
Can the proposed statement be a theorem?