The document defines key terms and concepts relating to matrices. It explains that a matrix is a two-dimensional array with rows and columns used to organize data. Matrices allow for simple display of data with non-essential information removed. They have various applications including graphic design and solving equations. The document defines order, elements, and notation for referring to entries in a matrix. It provides examples for stating the order, values, and positions of elements in matrices. Finally, it describes special types of matrices such as column, row, square, and triangular matrices as well as properties like the transpose and leading diagonal.

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