Rank of a matrix/system of equations, continued
Recall. The rank of a matrix A is the number of pivot columns of its equivalent reduced
row echelon form.
Theorem 1 (Theorem 1.35). Let A be a m × n coefficient matrix of a homogeneous system,
and suppose rank(A) = r. Then the solution has n − r parameters.
Proof.
Theorem 2 (Theorem 1.36). Let A be the m × (n + 1) augmented matrix of a consistent
system in n variables so that rank(A) = r. Then
1. The system has a unique solution if r = n.
2. The system has infinnite solutions if r < n.
Why don’t we consider the case where r > n?
1.2.7: An application to resistor networks
An important equation in current flow is
V = IR,
where potential V is in volts (V ), current I is in amperes (A), and resistance R is in ohms
(Ω).
Circuits
We can depict electric circuits schematically.
Power sources:
Resistors:
Theorem 3 (Kirchhoff’s Law). The sum of the voltage drops across the resistors (I · R)
clockwise around a circuit is equal to the voltage source
Example 1. Use Kirchhoff’s Law to calculate the current in the following circuit.
Example 2. Use Kirchhoff’s law to find the value of the indicated currents. Note that
whatever resistors are shared between two different loops have a net voltage drop based on
the (signed) sum of the currents.
Chapter 2: Matrices
Section 2.1: Matrix Arithmetic
Matrix arithmetic arises from systems of linear equations, and what would the necessary
operations look like so that we would arrive at a corresponding system of equations.
A m × n matrix is a rectangular array of numbers with m rows and n columns.
A square matrix is a matrix with the same number of rows as columns: an n × n matrix.
If A1 , A2 , . . . , An are the columns [column vectors] of the matrix A, then we often write
A = A1 A2 · · ·
An .
An n × 1 matrix is often called a column vector.
Example 3. Write the column vectors of the

1

B= 4
7
following matrix:

2 3
5 6
8 9
In a matrix A, the (i, j) entry, i.e. the row i and column j entry, is often notated aij . (In
a matrix B, the entries are bij , etc.)
Sometimes you’ll see the (i, j) entry of the matrix A is denoted Aij to emphasize that
the entry comes from matrix A. This notation is particularly useful is the matrix we are
referring to is the result of some operation on matrices (such as addition or multiplication
[see the near future]).
Example 4.


4 1 0 3
A = −1 7 −3 5
−4 2 −6 9
ˆ a12 =
ˆ a34 =
ˆ a32 =
ˆ a43 =
Definition 1. A m × n matrix is a zero matrix if every entry is 0.
Notation: 0
Two m × n matrices A and B are equal, or A = B, if A = [aij ], B = [bij ], and aij = bij
for every 1 ≤ i ≤ m and 1 ≤ j ≤ n.
Note. Two matrices of different sizes can NOT be equal.
Example 5 (Nonexample).


0 0
0 0 0
̸= 0 0
0 0 0
0 0
2.1.1: Matrix addition
Let A = [aij ] and B = [bij ] be two m × n matrices. Then A + B = C, where C is an m × n
matrix with aij + bij = cij for every i, j.
In other words, (A + B)ij = Aij + Bij .
The sum is computed componentwise.
Example 6.
1 2 3
−2 −1 1
+
=
4 0 −1
0
0 0
1 2 3
−2 −1 1
+
=
4 0 −1
1
2 3
Properties of addition
Let m and n be positive integers and A, B, and C be m × n matrices. Then the following
properties hold:
ˆ Commutativity: A + B = B + A
ˆ Associativity: (A + B) + C = A + (B + C)
ˆ Additive identity: A + 0 = A = 0 + A
ˆ Additive inverse: Given m × n matrix A, there exists a m × n matrix −A such that
A + (−A) = 0.
In particular, if A = [aij ], then −A = [−aij ].
Example 7. An example of additive inverses:
2.1.2: Scalar multiplication of matrices
Let A = [aij ] be a matrix and k a scalar (number). Then we define scalar multiplication by
kA = [k · aij ].
Example 8. Some examples of scalar multiplication:
1 3
2
=
5 7
0 2 −1
−3
=
−2 3 11
Properties of scalar multiplication
For any scalars r, s and m × n matrices A and B:
ˆ (r + s)A = rA + sA
ˆ r(A + B) = rA + rB
ˆ 1A = A
ˆ r(sA) = (rs)A
2.1.3: Matrix multiplication
In R1 a linear equation is of the form ax = b.
If we have m equations and n unknowns, the system of linear equations is of the form
a11 x1 + a12 x2 + · · · + a1n xn = b1
a21 x1 + a22 x2 + · · · + a2n xn = b2
..
.
am1 x1 + am2 xn + · · · + amn xn = bm .
The coefficient matrix is

a11
 a21

A =  ..
 .
···
···
...
a12
a22
am1 am2 · · ·
The vector of variables is


x1
 x2 
 
⃗x =  ..  ,
.
xn
and the vector of constants is


b1
 b2 

⃗b = 
 .. 
 . 
bm

a1n
a2n 

..  .
. 
amn
Q: How do we define multiplcation so that our system can be represented as
A⃗x = ⃗b?

a11 x1 + a12 x2 + · · · + a1n xn
a21 x1 + a22 x2 + · · · + a2n xn
..
.




am1 x1 + am2 x2 + · · · + amn xn



b1
  b2 
  
 =  .. 
  . 
bm
i.e. we want

a11
 a21

 ..
 .
a12
a22
···
···
..
.
am1 am2 · · ·
  
a11 x1 + a12 x2 + · · · + a1n xn
a1n
x1




a2n   x2   a21 x1 + a22 x2 + · · · + a2n xn
..
..   ..  = 
.
.  .  
amn
xn
am1 x1 + am2 x2 + · · · + amn xn





Since

a11 x1 + a12 x2 + · · · + a1n xn
a21 x1 + a22 x2 + · · · + a2n xn
..
.




am1 x1 + am2 x2 + · · · + amn xn







a11
a12
a1n
  a21 
 a22 
 a2n 
 





 =  ..  x1 +  ..  x2 + · · · +  ..  xn ,
  . 
 . 
 . 
am1
am2
amn
We can perform a multiplication using combinations of addition and multiplication:

a11
 a21

 ..
 .
a12
a22
···
···
..
.
am1 am2 · · ·

a1n
a2n 

.. 
. 
amn

 





x1
a11
a12
a1n
 x2   a21 
 a22 
 a2n 
  





 ..  =  ..  x1 +  ..  x2 + · · · +  ..  xn ,
.  . 
 . 
 . 
xn
am1
am2
amn
(m × n matrix )(n × 1 matrix ) = (m × 1 matrix )
In the product AB, the number of columns of A must be equal to the number of rows
of B. For each entry in the product, the entries in the corresponding row of the first matrix
are multiplied by the entries in the corresponding column of the second matrix. Generally,
we have
(m × p matrix )(p × n matrix ) = (m × n matrix )
Example 9. A 2 × 3 times 3 × 1 example:


5
7
11
3
1
0


1





 2  =




3