Overview of Section 2.1
This section introduces matrix multiplication.
• Matrices satisfy the same properties as vectors under addition:
A+B =B+A
(A + B) + C = A + (B + C) A + 0 = A
r(A + B) = rA + rB
(r + s)A = rA + sA
r(sA) = (rs)A
• We have seen how to multiply Ax, where A is a m × n matrix and x ∈ Rn . Let B =
[v1 v2 . . . vk ] be a n × k matrix, where the vi ∈ Rn are column vectors. Then AB is defined
to be the m × k matrix [Av1 Av2 . . . Avk ].
• The product AB can be computed using the above formula or component by component.
For example, one can compute the ijth entry of AB without computing any other entries.
• Under matrix multiplication, the following properties hold for a m×n matrix and appropriate
matrices B, C:
A(BC) = (AB)C
associative law
A(B + C) = AB + AC
left distributive law
(B + C)A = BA + CA
right distributive law
r(AB) = (rA)B = A(rB)
associativity and commutative law
for scalar multiplication r
Im A = A = AIn
identity law
n
• A = A · A · A · · · · A with n terms on the right.
• Commutativity does not hold for matrix multiplication. AB 6= BA in general. For some
matrices it does (such as IA = AI and A · A2 = A2 · A), but in general it doesn’t.
• Transpose of a matrix satisfies (AT )T = A, (A + B)T = AT + B T , (rA)T = rAT , and
(AB)T = B T AT . Note that the multiplication is reversed here!
Overview of Section 2.2
This section introduces the inverse of a matrix.
• A real number a 6= 0 has a multiplicative inverse b. The inverse b has the property that
a · b = 1 and b · a = 1. The number a has only one multiplicative inverse and we denote it
using the symbol a−1 . The formula for the inverse is a−1 = a1 .
• We define the multiplicative inverse for a matrix A in the same manner. B is a multiplicative
inverse to A if A · B = I and B · A = I. There is only one multiplicative inverse for the
matrix A and we denote it by the symbol A−1 . Unlike for real numbers, there is no (easy)
general formula for!A−1 .
a b
• When A =
is a 2 × 2 matrix, we have A =
c d
• Only square n × n matrices A have an inverse.
1
det A
d
−c
!
−b
a
.
• Thm: (Uniqueness of Inverses) If AB = BA = I and AC = CA = I, then B = C.
• If A is an invertible n × n matrix, then the equation Ax = b has the solution x = A−1 b. The
solution is unique.
1
• Thm: Assume A, B are invertible matrices. Then
a. A−1 is invertible and (A−1 )−1 = A
b. AB is invertible and (AB)−1 = B −1 A−1
c. AT is invertible and (AT )−1 = (A−1 )T .
The proof of this theorem is important and uses the theorem that inverses are unique. Be
sure to understand the proof.
• The product of any number of invertible matrices is an invertible matrix. One has
(AB . . . Z)−1 = Z −1 · · · B −1 A−1 .
• There are certain simple matrices called elementary matrices with an important property.
Suppose we perform a single row operation to the matrix A to form B. Then there is an
elementary matrix E with the property that B = EA. The elementary matrices E are
invertible.
• Thm: Let A be a n × n matrix. Then A is invertible ⇔ the matrix (A I) can be row reduced
to the matrix (I B). If this can be done, B = A−1 .
Overview of Section 2.3
This section reviews the many different properties that an invertible matrix can have.
• The main theorem is Thm 8 (The Invertible Matrix Theorem). Be sure to understand how
to use the theorem’s power. If one has a problem, one can often reduce a hypothesis to one
of the 12 statements (a)-(l). Then one can conclude that all the others are true. One then
uses whichever of these parts is most useful.
• Thm: If A, B are square matrices with AB = I, then A, B are both invertible and B = A−1 .
Note that this theorem does not follow directly from the definition of an inverse in Section
2.2. It follows from the Invertible Matrix Theorem
• If T : Rm → Rn is a linear transformation, it is said to be invertible if there is a linear
transformation S : Rn → Rm with the property that S ◦ T = Idm , and T ◦ S = Idn . This
definition is analogous to the definition for invertible matrices.
• As with matrices, an invertible linear transformation T has only one inverse and we denote
it by T −1 (x).
• Thm: If T is a linear transformation, then T is invertible ⇔ T is onto and one-to-one.
• The only invertible linear transformation have the domain equal to the codomain.
• Thm: Let T : Rn → Rn be a linear transformation with matrix T (x) = Ax. Then
a. T is invertible ⇔ A is an invertible matrix.
b. When (a) holds true, then the inverse S is given by the formula T −1 (x) = A−1 x.