3. INTRODUCTION
• Definition
A matrix is a rectangular array of elements or
entries aij involving m rows and n columns
nmij
ijiii
j
j
j
a
aaaa
aaaa
aaaa
aaaa
A
321
3333231
2232221
1131211 Rows, m
Columns, n
4. INTRODUCTION
• Definition
i. 2 matrices and
are said to be equal iff m = r and n = s then A
= B.
ii. If aij for i = j, then the entries a11,a22,a33,…
are called the diagonal of matrix A
nmij MaA nmij MbB
5. TYPES OF MATRICES
Square Matrix
Matrix with order n x n
33
987
654
321
43
21
BA
nm
6. TYPES OF MATRICES
Diagonal Matrix
Matrix with order n x n with aij ≠ 0 and aij = 0
for i ≠ j
200
010
001
A
7. TYPES OF MATRICES
Scalar Matrix
A diagonal matrix in which the diagonal
elements are equal, aii = k and aij = 0 for i ≠ j
where k is a scalar
2,
100
010
001
2
200
020
002
kA
8. TYPES OF MATRICES
Identity Matrix
A diagonal matrix in which the diagonal
elements are ‘1’, aii = 1 and aij ≠ 0 for i ≠ j
10
01
100
010
001
BA
9. TYPES OF MATRICES
Zero Matrix
A matrix which contains only zero elements,
aij = 0
00
00
000
000
000
BA
10. TYPES OF MATRICES
Negative Matrix
A negative matrix of A =[aij] denoted by –A
where -A =[-aij]
164
300
201
164
300
201
AA
11. TYPES OF MATRICES
Upper Triangular Matrix
If every elements below the diagonal is zero or
aij = 0, i > j
100
310
221
A
DIAGONAL
12. TYPES OF MATRICES
Lower Triangular Matrix
If every elements above the diagonal is zero or
aij = 0, i < j
143
012
001
A
DIAGONAL
13. TYPES OF MATRICES
Transpose of Matrix
If A =[aij] is an m x n matrix, then the
transpose of A, AT =[aij]T is the n x m matrix
defined by [aij] = [aji]T
152
411
321
143
512
211
T
AA
14. TYPES OF MATRICES
Properties Transposition Operation
• Let A and B matrices and k, . Then,
TTT
TT
TT
BABAiii
kAkAii
AAi
)
)
)
Rk
15. TYPES OF MATRICES
Example 1:
• If and , find
T
T
T
ABiii
Bii
Ai
)
2)
)
254
321
A
2
3
1
B
16. TYPES OF MATRICES
Answer 1:
231)
4622)
2
5
4
3
2
1
)
T
T
T
ABiii
Bii
Ai
17. TYPES OF MATRICES
Symmetric Matrix
If AT = A, where the elements obey the rule
aij = aji
324
221
415
324
221
415
T
AA
18. TYPES OF MATRICES
Skew Symmetric Matrix
If AT = - A, where the elements obey the rule
aij = - aji, so that the diagonal must contain
zeroes.
074
701
410
A
20. TYPES OF MATRICES
Row Echelon Form (REF)
Matrix A is said to be in REF if it satisfies the
following properties:
• Rows consisting entirely zeroes occur at the bottom of the
matrix.
• For each row that doesn’t consist entirely of zeroes, the 1st
nonzero is 1.
• For each non zero row, number 1 appear to the right of the
leading 1 of the previous row.
22. TYPES OF MATRICES
Reduced Row Echelon Form (RREF)
Matrix A is said to be in REF if it satisfies the
following properties:
• Rows consisting entirely zeroes occur at the bottom of the
matrix.
• For each row that doesn’t consist entirely of zeroes, the 1st
nonzero is 1.
• For each non zero row, number 1 appear to the right of the
leading 1 of the previous row.
• If a column contains a leading 1, then all other entries in
the column are zero
24. Types of Solutions
Consistent System
One solution
Consistent System
Infinite solutions
Inconsistent System
No solution
25. A linear equation is an equation that can be written in the form:
The coefficients ai and the constant b can be real or complex numbers.
A Linear System is a collection of one or more linear equations in the
same variables. Here are a few examples of linear systems:
bxaxaxa nn2211
4z2y2x
2zy3x2
1zyx
4xx4x
2xxx2x
1xx2x
532
5432
421
1zy2x
7z7y4x3
Any system of linear equations can be put into matrix form: bxA
The matrix A contains the coefficients of the variables, and the vector x has
the variables as its components.
For example, for the first system above the matrix version would be:
bxA
4
2
1
z
y
x
221
132
111
26. Prepared by Vince Zaccone
For Campus Learning Assistance
Services at UCSB
Here is the next system. The basic pattern is to start at the upper left corner, then use row
operations to get zeroes below, then work counterclockwise until the matrix is in REF.
3
0
1
310
310
111
RRR
R2RR
4
2
1
221
132
111
4z2y2x
2zy3x2
1zyx
13
*
3
12
*
2
3
0
1
000
310
111
RRR3
0
1
310
310
111
23
*
3
At this point you might notice a problem. That last row doesn’t make sense. It might help to
write out the equation that the last row represents.
It says 0x+0y+0z=3.
Are there any values of x, y and z that make this equation work? (the answer is NO!)
This system is called INCONSISTENT because we arrive at a contradiction during the
solution procedure. This means that the system has no solution.
27. This line is the
intersection of a pair of
the planes
This line is the intersection
of a different pair of the
planes
Prepared by Vince Zaccone
For Campus Learning Assistance
Services at UCSB
No Solution
If the system is inconsistent there will be no solutions.
In this case there will be a contradiction that appears
during the solution process.
This is the reduced matrix (actually we could go one step further and get a
zero up in row 1). Notice that we got a row of zeroes in the left part of the
augmented matrix. When this happens the system will either be inconsistent,
like this one, or we will have a free variable (infinite # of solutions).
4z2y2x
2zy3x2
1zyx
3
0
1
000
310
111
29. Inverse Matrix Method
Given a set of linear equations
x + 2y = 4
3x − 5y = 1
It can be written in matrix form as…
=
AX = B
This is the matrix form of the linear equations
1 2
3 −5( ) x
y( ) 4
1( )
30. Given that…
AX = B
A-1AX = A-1B
(A−1A = I, the identity matrix and multiplying any matrix
by I leaves the matrix unchanged)
X = A−1B
