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Lecture 6
Andrei Antonenko
February 12, 2003
1 Multiplicative inverse
In this lecture we will continue with the properties of matrix operations.
(M4) Existence of the multiplicative inverse. Matrix B is called the inverse for the
square matrix A if BA = AB = I. The existence of the inverse is a very difficult
question, which we will solve later. Now we’ll simply give some examples, and then a
formula for an inverse of 2 × 2-matrices.
Example 1.1. The matrix
doesn’t have an inverse, since if
a b
c d
is an inverse,
than
a b
c d
, from what
a b
which is never the case.
Example 1.2. The inverse for the matrix
is
since
Now we’re ready to give a formula for an inverse of 2 × 2-matrix.
Proposition 1.3. The inverse of 2 × 2 -matrix
a b
c d
exists if and only if ad − bc 6 = 0 and
a b
c d
ad − bc
d −b
−c a
Proof. If ad − bc 6 = 0 then we can simply check that this matrix is the inverse:
(
a b
c d
×
ad − bc
d −b
−c a
ad − bc
ad − bc −ab + ab
cd − cd −cb + da
We will not give a proof that if a matrix has an inverse, then ad − bc 6 = 0. This fact can be
generalized to the case of larger matrices, and we’ll prove it later in more general form.
The proposition above is useful when you want to get an inverse of a 2 × 2-matrix. Later
we’ll provide a method of finding the inverses of larger matrices, but for 2 × 2-matrices this is
the easiest one.
2 Transpose of a matrix
Definition 2.1. Matrix B is called transpose of a matrix A (notation: B = A
) if bij = aji.
In other words, we should take rows of a matrix A and write them as columns of matrix B.
Then B = A
. In general form, if
A =
a 11 a 12 · · · a 1 n
a 21 a 22 · · · a 2 n
. . .
am 1 am 2 · · · amn
then
A
=
a 11 a 21 · · · am 1
a 12 a 22 · · · am 2
. . .
a 1 n a 2 n · · · amn
Let’s notice, that if A is an m × n-matrix, then A
is an n × m-matrix.
Example 2.2. Let A =
. Than A
=
Example 2.3. Let A =
. Than A
=
. In general,
a 1 a 2 · · · an
a 1
a 2
. . .
an
, and
a 1
a 2
. . .
an
a 1 a 2 · · · an
Then with this system we can associate an m × (n + 1)-matrix
a 11 a 12 · · · a 1 n b 1
a 21 a 22 · · · a 2 n b 2
. . .
am 1 am 2 · · · amn bm
As for systems, for matrices we can define elementary operations, REF, RREF, and use the
same algorithms as for systems to reduce matrices to REF and RREF. So, given a system, we
can write its matrix, and then perform all the operations to transpose it to some of these forms,
and then get back to the system and write the solution for it.
Let’s consider a linear system:
a 11 x 1 + a 12 x 2 + · · · + a 1 nxn = b 1
a 21 x 1 + a 22 x 2 + · · · + a 2 nxn = b 2
am 1 x 1 + am 2 x 2 + · · · + amnxn = bm
We can write 3 following matrices associated with this system:
A =
a 11 a 12 · · · a 1 n
a 21 a 22 · · · a 2 n
. . .
am 1 am 2 · · · amn
, B =
b 1
b 2
. . .
bm
, X =
x 1
x 2
. . .
xn
Now we can see that the system (1) can be written in the following form:
AX = B (3)
Let’s check it:
AX = B ⇐⇒
a 11 a 12 · · · a 1 n
a 21 a 22 · · · a 2 n
. . .
am 1 am 2 · · · amn
x 1
x 2
. . .
xn
b 1
b 2
. . .
bm
a 11 · x 1 + a 12 · x 2 + · · · + a 1 n · xn
a 21 · x 1 + a 22 · x 2 + · · · + a 2 n · xn
am 1 · x 1 + am 2 · x 2 + · · · + amn · xn
b 1
b 2
. . .
bm
a 11 x 1 + a 12 x 2 + · · · + a 1 nxn = b 1
a 21 x 1 + a 22 x 2 + · · · + a 2 nxn = b 2
am 1 x 1 + am 2 x 2 + · · · + amnxn = bm
4 Elementary row operations in matrix form
Now, we’ll consider elementary row operations in matrix form.
4.1 Interchanging
Let’s imagine that we want to interchange two rows if a matrix. It can be done by multiplying
by the appropriate matrix. So, if we want to interchange rows i and j then we should use the
following matrix:
i
j
= Pij
where all the elements that are not shown are the same as the elements of the identity matrix.
If we multiply this matrix by A, i.e. take Pij A, we’ll get:
Pij A =
a 11 a 12 · · · a 1 n
aj 1 aj 2 · · · ajn
ai 1 ai 2 · · · ain
am 1 am 2 · · · amn
for A =
a 11 a 12 · · · a 1 n
ai 1 ai 2 · · · ain
aj 1 aj 2 · · · ajn
am 1 am 2 · · · amn
4.2 Multiplication
Let’s imagine that we want to multiply a row of a matrix by a given number c 6 = 0. It can be
done by multiplying by the appropriate matrix. So, if we want to multiply row i by c 6 = 0 then
we should use the following matrix:
i
...... c......
. . .
= Qi(c)
We will do it using matrix P 23 :
Example 4.2. Let’s suppose we want to multiply the 2nd row of the matrix
by 4. We will do it using matrix Q 2 (4):
Example 4.3. Let’s suppose we want to add the 1st row of the matrix
multiplied by 3 to the 3rd row. We will do it using matrix I + I 31 :
So, as we can see, all elementary row operations can be considered as a multiplication by
the appropriate matrix.
