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EECM 3714

Lecture 12: Unit 12

Matrix algebra
Renshaw, Ch
...
19
• Definitions, notation
• Matrix operations (Transposition – page 620; Addition; subtraction; Scalar multiplication; Matrix
multiplication - Page 621-3)
• Determinants

• Matrix inversion
• 2 by 2 inversion

• 3 by 3 inversion
• Solving systems of linear equations (Matrix Inversion and Cramer’s rule)

DEFINITIONS, NOTATION
• Matrix is a rectangular array of numbers/variables, e
...
(Page 578-579):

• 𝐴3×3

𝑎
= 𝑑
𝑔

𝑏
𝑒
ℎ

𝑐
𝑓 , 𝐵2×2 = 1 2
3 0
𝑖

• Order = dimensions of a matrix

• Order = number of rows (r) by number of columns (c)
• Usually denoted as m n, m = rows, n = columns

• An element is an entry in a matrix, denoted as 𝑎𝑖𝑗 , e
...
the element 𝑎23 = 𝑓 in matrix A, while
the element 𝑏22 = 0 in matrix B

SPECIAL MATRICES
• Square matrix: number of rows = number of
columns, i
...
𝑚 = 𝑛
1 3 5
• E
...
𝐶 = 7 6 4
0 23 1
• Null matrix: every element of matrix = 0 e
...

0 0
0=
0 0
• Identity matrix: diagonal elements are all 1; all
other elements are 0
1 0
• Note: must be a square matrix, e
...
𝐼 =
0 1

VECTORS AND SCALARS
• Scalar is a 1 × 1 matrix, i
...
a constant

• Row vector: matrix with only one row,
i
...
𝑚 = 1, e
...
𝑅 = 1 5 2
• Column vector = matrix with only one
2
column, i
...
𝑛 = 1, e
...
𝐷 = 4
1

EQUALITY OF TWO MATRICES
Two matrices A and B are equal if and only if
1) they have the same order and
2) if every element in A is equal to the corresponding element in B, e
...

𝐴=

2
1

5
2
,𝐵 =
2
1

5
⟹𝐴=𝐵
2

TRANSPOSITION
• Transposition involves interchanging the row and column entries of a matrix
• Notation: If A is a matrix with m rows and n columns, then its transpose, denoted by 𝐴𝑇 = 𝐴′ has
n rows and m columns
1 3 4
• Suppose that 𝐴 = 𝑎 𝑏 𝑐 and 𝐵 =
6 2 5
𝑎
1 6
• The transposes are then 𝐴′ = 𝑏 and 𝐵′ = 3 2
𝑐
4 5
• Note how the first row becomes the first column, the second row becomes the second column,
etc
...
e
...
e
...
)

0 9
6 7
3
;𝐵 =
;𝐶 =
1 50
0 4
10
• Note that A and B are conformable for addition/subtraction, while A and C; B and C are not
...

𝑎 𝑏
• Suppose that 𝐴 =

...

𝑐 𝑑
𝑘𝑎 𝑘𝑏
• 𝑘𝐴 =
𝑘𝑐 𝑘𝑑

MATRIX MULTIPLICATION, 1
• Before multiplying two matrices, first ensure that they are conformable for multiplication

• This involves checking if the number of columns of the first matrix = number of rows of second
matrix
• The order of the new matrix is given by the number of rows of the first matrix and the number of
columns of the second matrix

• Suppose A is a 2 × 3 matrix and B is a 3 × 2 matrix
...
Because B has 2 columns while A has 2 rows

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