Introduction to matrices and vocabulary

2. Learning Intention and Success
Criteria
 Learning Intention: Students will understand the
meaning of a variety of terms relating to matrices
 Success Criteria: You can use accurate mathematical
terminology to describe matrices and correctly
identify characteristics of a variety of matrices.

3. What is a Matrix?
 Matrix: A two-
dimensional array with
rows and columns used
to organise data.
 Denoted with a capital
letter
 Has square brackets
around it
 The data is often
numerical, but does not
have to be

4. Why Matrices?
 Simply put, matrices are
a simple way of
displaying data, with
extraneous information
removed
 Other applications:
 Graphic design (like
reflections and
shadows)
 Solving Equations

5. Vocabulary
 Order: The dimension of
the matrix "row x column"
m x n
 The order of matrix 𝐴 is
2 × 3 because there are 2
rows and 3 columns
 𝑚: The number of rows
(horizontal)
 𝑛: The number of columns
(vertical)
 (Do not use M or N as
names of matrices)
 Element: the term for each
number in a matrix
 Referring to entries in a
matrix:
 Lowercase letter (the same
letter as the name of the
matrix)
 Followed by numbers for
row,column
 Ex: 𝑎1,2 would be the entry
in the first row and the
second column of the
matrix 𝐴.

6. Example #1
For the matrix 𝐴 =
2 −5 3
4 0 8
,
a) State the order
 Matrix 𝐴 has order 2 × 3
b) State the name of the matrix
 𝐴
c) State the values of the elements in position:
i. 𝑎2,1
ii. 𝑎1,3
d) State the position of the element -5.
 1st
row, 2nd
column: the position is 𝑎1,2
 2nd
row, 1st
column = 4
 1st
row, 3rd
column = 3

7. Example #2
B is a 2 by 3 matrix. The elements of the matrix are
defined according to the rule 𝑏𝑖𝑗 = 𝑖 − 𝑗. What is the
matrix?
𝑏1,1 = 1 − 1 = 0
𝑏2,1 = 2 − 1 = 1
Calculate this for each element of 𝐵
𝐵 =
0 −1 −2
1 0 −1

8. Special Types of Matrices
 Column Matrix: has exactly one column. Dimension is
𝑚 × 1
 Row Matrix: has exactly one row. Dimension is 1 × 𝑛
 Square matrix: has the same number of rows and
columns. Dimension is 𝑛 × 𝑛
 Transpose of a Matrix: Given a matrix 𝐴, the transpose
of A (denoted 𝐴 𝑇) swaps the rows and columns
 Entry 𝑎1,2 in matrix 𝐴 will become entry 𝑎2,1
𝑇
in matrix 𝐴 𝑇
 If 𝐴 has dimension 𝑚 × 𝑛, then 𝐴 𝑇
has dimensions 𝑛 × 𝑚
 E.g. If 𝐴 =
1 3 5
−7 −8 −9
, then 𝐴 𝑇
=
1 −7
3 −8
5 −9

9. Square Matrices Sub-categories
 Leading diagonal: The entries
going from the top left to the
bottom right of a square matrix
 Diagonal Matrix: All entries not on
the leading diagonal must be zero.
 𝐵 =
3 0 0
0 5 0
0 0 −4
 Identity Matrix (named I): A
diagonal matrix where the leading
diagonal is filled with 1s
 𝐼 =
1 0 0
0 1 0
0 0 1
Triangular Matrix
 Upper-triangular: when all entries
below the leading diagonal are zero.

1 2 0
0 3 4
0 0 5
 Lower-triangular: when all entries
above the leading diagonal are zero.

1 0 0
0 −4 0
3 7 1
 Note that the all diagonal matrices
are both upper and lower-
triangular