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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
...
621
• Multiply the first element of the first row with the first element of the first column; then multiply the second
element of the first row with the second element of the first column, etc
...


• Do the same with all row and column combinations (sum of the products of elements of row 1 and column 2
will yield the 1,2 element in the product matrix, etc
...
Find AB
3 4
2
A is a 2 by 2 matrix; B is a 2 by 1 column vector
...
The resulting matrix is a 2 by 1
column vector…
(1 × 1) + (2 × 2)
1 2 1
5
𝐴𝐵 =
=
=
(3 × 1) + (4 × 2)
3 4 2
11
2 4
1 3
If 𝐶 =
;𝐷 =
, find C-D
6 8
5 7
Note that C and D are conformable for subtraction (same order, 2 by 2)
2 4
1 3
1 1
𝐶−𝐷 =
−
=
6 8
5 7
1 1
Do examples 19
...


• Suppose that 𝐴 =
•

•
•
•
•

•

Examples
𝑎
• If 𝐸 = 𝑐
𝑒

𝑏
𝑑 , find E’
...
find 100𝑍
...


𝐴𝑛𝑠: 100𝑍 = 100 100

1000 ∴ 100𝑍 = 10000

100000

1 2 3
1
• If 𝑋 = 4 5 6 , 𝑌 = 2 , find XY
...
e
...
e
...

𝑎 𝑏
, 𝑡ℎ𝑒𝑛 𝐷𝑒𝑡 𝐴 = 𝑎𝑑 − 𝑏𝑐
𝑐 𝑑
• For a 3 × 3 matrix, we can use Sarrus’ rule (ONLY works for 3 by 3 matrices!!): If
• For a 2 × 2 matrix: If 𝐴 =

+
𝑐
𝑎
𝑓 , 𝐷𝑒𝑡 𝐴 =
𝑑
𝑖
𝑔
• Then, 𝐷𝑒𝑡 𝐴 = 𝑎𝑒𝑖 + 𝑏𝑓𝑔 + 𝑐𝑑ℎ − 𝑎𝑓ℎ − 𝑏𝑑𝑖 − 𝑐𝑒𝑔
𝑎
𝐴= 𝑑
𝑔

𝑏
𝑒
ℎ

+ +
𝑏 𝑐
𝑒 𝑓
ℎ 𝑖

− −
𝑎 𝑏
𝑑 𝑒
𝑔 ℎ

−
𝑐
𝑓
𝑖

EXAMPLES
2 4
Find the determinants of the following matrices • 𝐵 = 4 3
4 −7
2 1
• 𝐴=
1 3
+
• 𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
2
𝐷𝑒𝑡 𝐵 =
4
𝐷𝑒𝑡 𝐴 = (4 × 3) − −7 × 1
...

𝐷𝑒𝑡 𝐴 = 19
...

𝑏

• Normally, we’d solve this as 𝑥 = 𝐴
...
But
...


• Therefore, we can solve for x as follows : 𝑥 = 𝐴−1 𝑏
• So we can solve for a system if the matrix A has an inverse

• Only square matrices have inverses; only non-singular matrices have inverses
...
Find the inverse of the matrix A as follows:
𝑐 𝑑
• First, find 𝐷𝑒𝑡(𝐴)
...

𝐷𝑒𝑡(𝐴) −𝑐
𝑎𝑑−𝑏𝑐
𝑎
−𝑐 𝑎
𝐴−1 =

𝑑
𝑎𝑑−𝑏𝑐
−𝑐
𝑎𝑑−𝑏𝑐

−𝑏
𝑎𝑑−𝑏𝑐
𝑎
𝑎𝑑−𝑏𝑐


...
6 & 19
...


INVERSE OF 3 BY 3 MATRIX, 1
• The inverse of A is then
𝐴−1 =

1
𝐴𝑑𝑗(𝐴)
...
8

− 𝑀21
𝑀22
− 𝑀23
𝐶31
𝐷𝑒𝑡(𝐴)
𝐶32
𝐷𝑒𝑡(𝐴)
𝐶33
𝐷𝑒𝑡(𝐴)

1
𝐷𝑒𝑡(𝐴)

𝑀31
− 𝑀32
𝑀33

𝐶11
𝐶12
𝐶13

𝐶21
𝐶22
𝐶23

𝐶31
𝐶32
...


EXAMPLE 2
2
• Find the inverse of 𝐵 = 4
2
• Solution:
• 𝐵−1 =

1
𝐴𝑑𝑗(𝐵)
𝐷𝑒𝑡(𝐵)

• Recall that 𝐷𝑒𝑡(𝐵) = 10
• First, fin

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