2. Linear Transformation
• Matrix Algebra developed in relation to linear transformations such
as the following:
𝑎𝑥 + 𝑏𝑦 = 𝑋
𝑐𝑥 + 𝑑𝑦 = 𝑌
Where 𝑎, 𝑏, 𝑐 and 𝑑 are real numbers. This transformation introduces
a function(mapping) by which an ordered pair 𝑥, 𝑦 in 𝑥𝑜𝑦 plane
transformed (associated) to another ordered pair (𝑋, 𝑌) in 𝑋𝑂𝑌 plane.
𝒙 𝑿
𝒀𝒚
𝒐 𝑶
This linear transformation can
be done through the coefficients
of 𝑥 and 𝑦 . The square array
𝑎 𝑏
𝑐 𝑑
represents this
transformation which is one
among many other
transformations. Such an array
is called matrix.
3. Matrix Algebra
• Definition: A matrix is a rectangular or square array of
elements (usually numbers) arranged in rows and
columns.
• Matrices are usually shown by capital and bold letters
such as A, B, etc. Matrix A with 3 rows and 2 columns is
shown by 𝑨 𝟑×𝟐 and matrix B with m rows and n columns
is shown by 𝑩 𝒎×𝒏. Their elements are shown by small
letters with an index indicating the position of the
element in the matrix.
• 𝐴3×2 =
𝑎11
𝑎21
𝑎31
𝑎12
𝑎22
𝑎32
𝐵 𝑚×𝑛 =
𝑏11
𝑏21
⋮
𝑏 𝑚1
𝑏12
𝑏22
⋮
𝑏 𝑚2
𝑏13 …
𝑏23 …
⋮ ⋯
𝑏 𝑚3 ⋯
𝑏1𝑛
𝑏2𝑛
⋮
𝑏 𝑚𝑛
4. Matrix Algebra
• There are other ways of showing a matrix:
𝑩 = 𝑏𝑖𝑗 𝑚×𝑛
𝒐𝒓 𝑩 𝒎×𝒏
The Order of a Matrix:
• The size and the shape of a matrix is given by its order
which is the multiplication of number of rows and
number of columns.
• In the previous examples the order of A is 3 × 2 and the
order of B is 𝑚 × 𝑛.
• If 𝑚 = 𝑛 then the matrix is called a square matrix of order
𝑚 (𝑜𝑟 𝑛).
5. Vectors & Scalars
• A matrix with just one row or one column is called vector.
𝐴1×3 = 2 −10 3.5 is a row (horizontal) vector.
𝐵4×1 =
2
−1.65
7.2
5
is a column (vertical) vector.
• In matrix algebra any real number is called scalar. So, a
scalar in matrix algebra is a 1 × 1 matrix.
6. Types of Matrices
Null (zero) Matrix:
If all elements of a matrix is zero the matrix is called null or
zero matrix and it is shown by 𝟎 .
𝐴2×2 =
0 0
0 0
𝐶2×3 =
0 0 0
0 0 0
Diagonal Matrix:
A square matrix which have at least one nonzero element on
its main diagonal and zeros elsewhere is a diagonal matrix.
𝐴3×3 =
3 0 0
0 −1 0
0 0 2
Main Diagonal𝒊 = 𝒋 → 𝒂𝒊𝒋 ≠ 𝟎
𝒊 ≠ 𝒋 → 𝒂𝒊𝒋 = 𝟎
7. Types of Matrices
Identity (unit) Matrix:
A diagonal matrix whose all elements on the main diagonal
are equal to one is called identity or unit matrix. A unit
matrix is usually shown by letter I and its order.
𝐼2×2 = 𝐼2 =
1 0
0 1
𝐼3×3 = 𝐼3 =
1 0 0
0 1 0
0 0 1
Scalar Matrix:
In a diagonal matrix if all elements are equal the matrix is
called a scalar matrix.
𝐴3×3 =
3 0 0
0 3 0
0 0 3
8. Types of Matrices
Transpose Matrix:
For a matrix 𝑨 𝒎×𝒏 the transpose is defined as 𝑨′ 𝒏×𝒎 (in some books 𝑨 𝒏×𝒎
𝑻
)
where the rows and columns are interchanged.
𝐴2×4 =
1 4
3 −2
1
0
−3
1.2
→ 𝐴′4×2 =
1
4
3
−2
1
−3
0
1.2
• Transposed of a row vector is a column vector and vice versa.
𝑋3×1 =
1
5
4
→ 𝑋′1×3 = 1 5 4
Properties of Transpose Matrix:
By the definition of transpose matrix we can conclude 𝑨′ ′ = 𝑨.
By the definition, 𝑰′ = 𝑰. This property is true for all diagonal matrices.
For a square matrix 𝑨, if 𝑨′
= 𝑨 , then 𝑨 is a symmetric matrix.
1 0.5
0.5 3
𝒌𝑨 ′ = 𝒌𝑨′
9. Types of Matrices
Triangular Matrices:
If all elements above the main diagonal of a square matrix are
zero the matrix is called “lower triangular matrix”.
e.g. 𝐴 =
2 0 0
0 −1 0
4 3 5
if 𝑖 < 𝑗 , 𝑎𝑖𝑗 = 0
Alternatively, If all elements under the main diagonal of a
square matrix are zero the matrix is called “upper triangular
matrix”.
e.g. 𝐵 =
1 −3 1
2
0 4 7
0 0 −6
if 𝑖 > 𝑗 , 𝑎𝑖𝑗 = 0
10. Types of Matrices
Symmetric Matrix:
A square matrix is symmetric if 𝑨 = 𝑨′. This means that the
elements above the main diagonal in the matrix are the mirror
image of elements under the main diagonal (the main diagonal
works as a mirror)
𝐴3×3 =
3 1.2 2
1.2 −1 0
2 0 2
Equality in matrices:
• Two matrices 𝑨 and 𝑩 are equal if they have the same order
and their corresponding elements are equal.
𝑨 = 𝑩 ↔
𝑜𝑟𝑑𝑒𝑟 𝑨 = 𝑜𝑟𝑑𝑒𝑟(𝑩)
∀𝑖, 𝑗 → 𝑎𝑖𝑗 = 𝑏𝑖𝑗
11. Matrix Operation
Scalar Multiplication:
If 𝑘 is a scalar then
𝑘. 𝑨 = 𝑘. 𝑎𝑖𝑗 𝑚×𝑛
This means that all elements of the matrix are multiplied by
the scalar 𝑘.
Matrix Addition & Subtraction:
Addition and subtraction are defined for the matrices of the
same order. It is not possible to add or subtract matrices
from different orders. In both cases the corresponding
elements are added or subtracted:
𝑨 𝒎×𝒏 ± 𝑩 𝒎×𝒏 = 𝑎𝑖𝑗 ± 𝑏𝑖𝑗 𝑚×𝑛
12. Matrix Operations
e.g. 𝐴 =
3 1 −2
2 4 1
and 𝐵 =
7 −10 4
5 0 3
𝐴 + 𝐵 =
10 −9 2
7 4 4
𝐴 − 𝐵 =
−4 11 −6
−3 4 −2
Properties of Addition & Subtraction:
𝑨 + 𝑩 = 𝑩 + 𝑨 Commutative law
𝑨 ± 𝑩 ± 𝑪 = 𝑨 ± 𝑩 ± 𝑪 Associative law
𝒌. 𝑨 ± 𝑩 = 𝒌𝑨 ± 𝒌𝑩 (𝒌 is a scalar)
𝑨 ± 𝑩 ′ = 𝑨′ ± 𝑩′ can be extended to “n”
matrices
13. Matrix Operations
• Matrix Multiplication:
Multiplication of two matrices 𝑨 and 𝑩, in the form of 𝑨 × 𝑩 or 𝑨𝑩, is
possible if the number of columns in 𝑨 is equal to the number of rows
in 𝑩. The result of this multiplication is another matrix 𝑪 where the
number of its rows is equal to the number of rows in 𝑨 and number of
its columns is equal to the number of columns in 𝑩; that is:
𝑨 𝒎×𝒏 × 𝑩 𝒏×𝒑 = 𝑪 𝒎×𝒑
Elements of 𝑪 can be calculated by adding some multiplications;
multiplications of the elements in the i-th row of 𝑨 by the
corresponding elements in the j-th column of 𝑩, that is:
𝑪𝒊𝒋 = 𝑘=1
𝑛
𝑎𝑖𝑘 𝑏 𝑘𝑗 where
𝑖 = 1,2, ⋯ , 𝑚
𝑗 = 1,2, ⋯ , 𝑝
14. Matrix Operations
• For example, matrix 𝑨 𝟑×𝟑 =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
cannot be multiplied by a
horizontal vector 𝑿 𝟏×𝟑 = 𝑥 𝑦 𝑧 but it can be multiplied by its
transpose which is a vertical vector; 𝑿′ 𝟑×𝟏 =
𝑥
𝑦
𝑧
and the result is:
AX =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝑥
𝑦
𝑧
=
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧
𝑑𝑥 + 𝑒𝑦 + 𝑓𝑧
𝑔𝑥 + ℎ𝑦 + 𝑖𝑧
• In the above example:
𝑿𝑿′
= 𝑥2
+ 𝑦2
+ 𝑧2
which is a scalar but 𝑿′
𝑿 =
𝑥2 𝑥𝑦 𝑥𝑧
𝑦𝑥 𝑦2 𝑦𝑧
𝑧𝑥 𝑧𝑦 𝑧2
which is a symmetric matrix, why?
15. Matrix Operations
Properties of Matrix Multiplication:
In general, 𝑨𝑩 ≠ 𝑩𝑨 if both exist, but there are special cases that
this property is not true.
If 𝑰 is an identity matrix 𝑰𝑩 = 𝑩𝑰 = 𝑩.
𝑨 𝑩 + 𝑪 = 𝑨𝑩 + 𝑨𝑪 and 𝑩 + 𝑪 𝑨 = 𝑩𝑨 + 𝑪𝑨
𝑨 𝑩𝑪 = 𝑨𝑩 𝑪
If 𝑨𝑩 exist then 𝑨𝑩 ′ = 𝑩′ 𝑨′ (this can be extended to more than 2
matrices, i.e.: 𝑨𝑩𝑪 ′
= 𝑪′𝑩′
𝑨′
From 𝑨𝑩 = 𝟎 we cannot conclude necessarily that 𝑨 = 𝟎𝑜𝑟 𝑩 = 𝟎.*
From 𝑨𝑩 = 𝑨𝑪 we cannot conclude necessarily that 𝑩 = 𝑪.**
16. Determinant of a Matrix
• Consider the system of simultaneous equations
𝒂𝒙 + 𝒃𝒚 = 𝒆
𝒄𝒙 + 𝒅𝒚 = 𝒇
Where 𝒂, 𝒃, … . , 𝒆, 𝒇 are constants of the system. If the coefficients of
𝒙 and 𝒚 in the first equation (i.e. 𝒂 and 𝒃 )have a linear relationship
with the coefficients of the second equation (i.e. 𝒄 and 𝒅 ), the system
either does not have a unique solutions for 𝒙 and 𝒚 (when 𝒆, 𝒇 also
have the same linear relationship) or there is no solution at all (the
system is not solvable as the equations are in contrary with each other).
• If
𝑎
𝑐
=
𝑏
𝑑
→ 𝑎𝑑 = 𝑏𝑐 or 𝑎𝑑 − 𝑏𝑐 = 0 it means the
coefficients have a linear relationship and there is no
unique solutions for 𝑥 and 𝑦. The value of 𝑎𝑑 − 𝑏𝑐
determines whether a system of simultaneous equations
have a unique solutions or not.
17. Determinant of a Matrix
o For the system of simultaneous equations A:
2𝑥 + 3𝑦 = 12
4𝑥 + 6𝑦 = 24
and
B:
2𝑥 + 3𝑦 = 12
4𝑥 + 6𝑦 = −18
we have:
2
4
=
3
6
→ 2 × 6 = 3 × 4 𝒐𝒓 2 × 6 − 3 × 4 = 0
So, both systems fail to provide unique
solutions for 𝑥 and 𝑦 but the difference
between them is that system A provides
infinite solutions (because there are, in fact,
one equation with two variables, which
geometrically means two lines coincide) but
the equations in system B are in contrary with
each other (geometrically means they are two
parallel lines and do not cross each other).
x
y
2𝑥 + 3𝑦 = 12
4𝑥 + 6𝑦 = 24
2𝑥 + 3𝑦 = 12
4𝑥 + 6𝑦 = −18
x
y
Infinite solutions
No solution
18. Determinant of a Matrix
• for matrix 𝑨 𝟐×𝟐 =
𝑎 𝑏
𝑐 𝑑
, the value of 𝑎𝑑 − 𝑏𝑐 is called
“determinant” of the matrix and it is shown by det 𝑨 or simply 𝑨 .
𝑨 𝟐×𝟐=
𝑎 𝑏
𝑐 𝑑
→ det 𝑨 = 𝑨 = 𝑎𝑑 − 𝑏𝑐
• To every square matrix we can correspond a scalar which is called the
determinant of the matrix. So, determinant of a matrix represents a
function.
• What about if the square matrix is 𝟑 × 𝟑 or even 𝒏 × 𝒏?
In order to obtain the determinant of matrices of higher orders than 2
we need to introduce two concepts:
Minors
Cofactors
19. Determinant of Matrices of Higher Orders than 2
• Minors: For every element (such as 𝑎𝑖𝑗) of a square matrix there
is a corresponding determinant, called “minor of 𝒂𝒊𝒋” (shown by
𝑀𝑖𝑗) derived from ignoring the elements in the same row and
column of 𝑎𝑖𝑗 (i.e. 𝑖 and 𝑗).
• For matrix
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
, minors are:
Minor of 𝑎11 = 𝑀11 =
𝑎22 𝑎23
𝑎32 𝑎33
= 𝑎22 𝑎33 − 𝑎23 𝑎32
Minor of 𝑎12 = 𝑀12 =
𝑎21 𝑎23
𝑎31 𝑎33
= 𝑎21 𝑎33 − 𝑎23 𝑎31
Minor of 𝑎13 = 𝑀13 =
𝑎21 𝑎22
𝑎31 𝑎32
= 𝑎21 𝑎32 − 𝑎22 𝑎31
Minor of 𝑎21 = 𝑀21 =
𝑎12 𝑎13
𝑎32 𝑎33
= 𝑎12 𝑎33 − 𝑎13 𝑎32
20. Determinant of Matrices of Higher Orders than 2
• Minor of 𝑎22 = 𝑀22 =
𝑎11 𝑎13
𝑎31 𝑎33
= 𝑎11 𝑎33 − 𝑎13 𝑎31
• ………………………………
• ………………………………
• Minor of 𝑎33 = 𝑀33 =
𝑎11 𝑎12
𝑎21 𝑎22
= 𝑎11 𝑎22 − 𝑎12 𝑎21
• Cofactors: Cofactors of each element 𝑎𝑖𝑗, shown by 𝐶𝑖𝑗, are minors with a
sign depending on the row and column of the element. i.e.:
𝐶𝑖𝑗 = −1 𝑖+𝑗 𝑀𝑖𝑗
So,
the cofactor of 𝑎11 is 𝑪 𝟏𝟏 = −1 1+1
𝑀11 = 𝑀11 = 𝑎22 𝑎33 − 𝑎23 𝑎32
And
the cofactor of 𝑎23 is
𝑪 𝟐𝟑 = −1 2+3 𝑀23 = −𝑀23= −(𝑎11 𝑎32 − 𝑎12 𝑎31) = −𝑎11 𝑎32 + 𝑎12 𝑎31
21. Determinant of Matrices of Higher Orders than 2
• The matrix of cofactors can be shown as:
𝐶 =
𝐶11 𝐶12 𝐶13
𝐶21 𝐶22 𝐶23
𝐶31 𝐶32 𝐶33
=
𝑀11 −𝑀12 𝑀13
−𝑀21 𝑀22 −𝑀23
𝑀31 −𝑀32 𝑀33
Now, we can define and calculate the determinant of a matrix with
order higher than two.
Definition: Determinant of a 𝑛 × 𝑛 matrix is the summation of
products between elements of any row (or any column ) and their
corresponding cofactors. i.e.:
For a matrix 𝑨 𝒏×𝒏 we can write:
𝑨 = 𝑎11. 𝑪 𝟏𝟏 + 𝑎12. 𝑪 𝟏𝟐 + ⋯ + 𝑎1𝑛 . 𝑪 𝟏𝒏 Based on the 1st row
𝑨 = 𝑎1𝑛. 𝑪 𝟏𝒏 + 𝑎2𝑛. 𝑪 𝟐𝒏 + ⋯ + 𝑎 𝑛𝑛 . 𝑪 𝒏𝒏 Based on the nth column
22. Determinant of Matrices of Higher Orders than 2
o Find the determinant of 𝐀 =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
.
Based on the elimination of rows and columns using the elements
of the first row we have:
𝑨 = 𝑎.
𝑒 𝑓
ℎ 𝑖
− 𝑏.
𝑑 𝑓
𝑔 𝑖
+ 𝑐.
𝑑 𝑒
𝑔 ℎ
= 𝑎 𝑒𝑖 − 𝑓ℎ − 𝑏 𝑑𝑖 − 𝑓𝑔 + 𝑐(𝑑ℎ − 𝑒𝑔)
= 𝑎𝑒𝑖 − 𝑎𝑓ℎ − 𝑏𝑑𝑖 + 𝑏𝑓𝑔 + 𝑐𝑑ℎ − 𝑐𝑒𝑔
o The determinant of the unit matrix of order 𝑛 is:
𝑰 𝒏×𝒏 = 𝑰 𝒏 =
1
0
0 …
1 ⋯
0
0
⋮ ⋮ ⋯ ⋮
0 0 … 1
𝑰 𝒏 = 𝑰 𝒏−𝟏 = ⋯ = 𝑰 𝟐 = 1 , why?
23. Sarrus’ Rule
• For a matrix 𝐀 =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
can be calculated through following steps:
1. Add the first 2 columns of the matrix to the right of the 3rd column:
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
2. Subtract the sum of the products along the green arrows from the sum of
products along the blue arrows:
𝑨 = 𝑎𝑒𝑖 + 𝑏𝑓𝑔 + 𝑐𝑑ℎ − (𝑐𝑒𝑔 + 𝑎𝑓ℎ + 𝑏𝑑𝑖)
• Note: It is also possible to add the first 2 rows of the matrix to the bottom of
the 3rd row:
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
(+) (-)
(+) (-)
𝑨 = 𝑎𝑒𝑖 + 𝑏𝑓𝑔 + 𝑐𝑑ℎ − (𝑐𝑒𝑔 + 𝑎𝑓ℎ + 𝑏𝑑𝑖)
24. Properties of Determinants
1) Transposing a matrix does not change its determinant: 𝑨 = 𝑨′
𝑎 𝑏
𝑐 𝑑
=
𝑎 𝑐
𝑏 𝑑
= 𝑎𝑑 − 𝑏𝑐
2) If all elements of a row (or column) of a square matrix are zero the
determinant of that matrix is zero. Why?
𝑎 0 2
𝑏 0 3
𝑐 0 4
= 0
3) If two rows (or columns) of a square matrix have the same values or
make a linear relationship with each other the determinant of the
matrix is zero.
𝒂 𝒃 𝒄
𝒂 𝒃 𝒄
𝑔 ℎ 𝑖
=
𝒂 𝒃 𝒄
𝟐𝒂 𝟐𝒃 𝟐𝒄
𝑔 ℎ 𝑖
= 0
25. Properties of Determinants
4) If the elements in a row (or in a column) of a square matrix
multiplied by a constant the determinant of the matrix is multiplied by
that constant but if the entire elements of a matrix multiplied by a
constant the determinant of the matrix multiplied by that constant to
the power of the order of the matrix, i.e.
If 𝐀 =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
then
𝑘. 𝑎 𝑏 𝑐
𝑘. 𝑑 𝑒 𝑓
𝑘. 𝑔 ℎ 𝑖
= 𝑘. 𝑨 and
𝑘. 𝑎 𝑘. 𝑏 𝑘. 𝑐
𝑘. 𝑑 𝑘. 𝑒 𝑘. 𝑓
𝑘. 𝑔 𝑘. ℎ 𝑘. 𝑖
=
𝑘3.
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝑜𝑟 𝑘. 𝑨 = 𝑘3. 𝑨
If matrix 𝑨 was from
order of 𝑛 then
𝑘. 𝑨 = 𝑘 𝑛. 𝑨
26. Properties of Determinants
5) For the square matrices 𝑨 and 𝑩 with the same orders
𝑨𝑩 = 𝑨 . 𝑩
6) If two rows (or two columns) of a square matrix are interchanged
the determinant of the matrix is multiplied by -1.
𝑎 𝑏
𝑐 𝑑
= −
𝑐 𝑑
𝑎 𝑏
𝑖𝑛𝑡𝑒𝑟𝑐ℎ𝑎𝑛𝑔𝑖𝑛𝑔 𝑡𝑤𝑜 𝑟𝑜𝑤𝑠
7) If the elements of a row (or a column) of a square matrix is the sum
of two row (column) vectors, the determinant of the matrix can be
written as the sum of two determinants; each corresponded to one of
the vectors, i.e.:
𝑎 + 𝜇 𝑏 + 𝜃
𝑐 𝑑
=
𝑎 𝑏
𝑐 𝑑
+
𝜇 𝜃
𝑐 𝑑
𝑎 + 𝜇 𝑏
𝑐 + 𝜃 𝑑
=
𝑎 𝑏
𝑐 𝑑
+
𝜇 𝑏
𝜃 𝑑
27. 8) Adding or subtracting a scalar multiple of a row (or a column) to
another row (column) does not change the determinant of the matrix.
𝑎 + 𝑘. 𝑏 𝑏
𝑐 + 𝑘. 𝑑 𝑑
=
𝑎 𝑏
𝑐 𝑑
+ 𝑘.
𝑏 𝑏
𝑑 𝑑
=
0
𝑎 𝑏
𝑐 𝑑
9) Determinant of a triangular, diagonal and scalar matrix is the
multiplication of the elements on the main diagonal.
Triangular matrix :
1 4 3
0 −2 5
0 0 3
= 1 × −2 × 3 = −6
Diagonal matrix:
1 0 0
0 −2 0
0 0 3
= 1 × −2 × 3 = −6
Scalar Matrix:
−2 0 0
0 −2 0
0 0 −2
= −2 × 𝐼3 = −2 3 × 𝑰 𝟑
1
= −8
Properties of Determinants
28. • The last two properties are sometimes used to facilitate the
calculation of determinant of a matrix.
o If 𝑨 =
2 3 −1
1 4 0
−3 5 4
find 𝑨 .
According to the property No. 8, if we substitute the last row (𝑅3) by
4𝑅1 + 𝑅3 (multiplying the first row by 4 and adding it to the third
row) the result of the determinant does not change. So:
2 3 −1
1 4 0
−3 5 4
=
2 3 −1
1 4 0
5 17 0
= −1 ×
1 4
5 17
= 3
• These type of operations are called elementary row/column
operations and they are useful to solve a system of simultaneous
equations . These types of operations will be discussed later.
Properties of Determinants
29. • The concept of inverse is very important in all branches of algebra.
Inverse of a real number, inverse of a function are just different aspects
of this concept.
• In matrix algebra the inverse of a square matrix 𝑨, which is shown by
𝑨−𝟏
(read 𝑨 inverse), is the matrix of the same order such that:
𝑨𝑨−𝟏
= 𝑨−𝟏
𝑨 = 𝑰
Where 𝑰 is an identity matrix of the same order.
Note: Not all square matrices have an invers but if a square matrix is
invertible, the inverse matrix is unique.
Some properties of inverse matrices are as following:
𝑨−𝟏 −𝟏
= 𝑨
𝑨𝑩 −𝟏
= 𝑩−𝟏
𝑨−𝟏
𝑨′ −𝟏
= 𝑨−𝟏 ′
𝑨𝑨−𝟏
= 𝑰 → 𝑨 . 𝑨−𝟏
= 1 → 𝑨−𝟏
=
1
𝑨
Invers of a Matrix
30. A square matrix 𝑨 is invertible if and only if 𝑨 ≠ 0. This is
necessary and sufficient condition for a square matrix to have an
inverse. If 𝑨 ≠ 0, the matrix is called non-singular and singular
otherwise.
To find the inverse of a function we can follow one of these
methods:
a) Using the Definition:
o Find the inverse of the matrix 𝑨 =
2 4
5 5
.
As 𝑨 = −10, so, the inverse exists. According to the definition, if
𝑨−𝟏
=
𝑎 𝑏
𝑐 𝑑
then : 𝐀𝑨−𝟏
=
2 4
5 5
𝑎 𝑏
𝑐 𝑑
=
1 0
0 1
= 𝑰. By
multiplication we have:
2𝑎 + 4𝑐 2𝑏 + 4𝑑
5𝑎 + 5𝑐 5𝑏 + 5𝑑
=
1 0
0 1
By solving the system of four simultaneous equations with four
variables we will have : 𝑎 = −0.5 , 𝑏 = −0.5 , 𝑐 = 0.5 and 𝑑 = −0.5.
Finding the Inverse of a Square Matrix
31. So, 𝑨−𝟏 =
−0.5 −0.5
0.5 −0.5
. This method can be difficult for matrices of
orders bigger than two.
b) Gauss Method (Gaussian Elimination Method):
A prerequisite for using this method is to know the concept of
elementary raw (column) operations. If a matrix is associated to a
system of simultaneous linear equations (called coefficients matrix)
elementary raw (column)operations help to solve the system and find
the set of solutions easily. They can be also used to calculate the
determinant of a square matrix or to find its inverse, in case the
matrix is invertible.
Three types of these operations are:
I. Row (column) Switching: A row (column) in a matrix can be
switched with another row (column), i.e. 𝑅𝑖 ↔ 𝑅𝑗 (𝐶𝑖 ↔ 𝐶𝑗)
Finding the Inverse of a Square Matrix
32. II. Row (column) Multiplication: all elements in a row (column) can
be multiplied by a non-zero scalar and be replaced by that, i.e.
𝑘. 𝑅𝑖 → 𝑅𝑖 (𝑘. 𝐶𝑖 → 𝐶𝑖)
III. Row (column) Addition/Subtraction: A row (column) can be
replaced by the sum of that row (column) and a multiple of
another row (column), i.e. 𝑅𝑖 ± 𝑘. 𝑅𝑗 → 𝑅𝑖 (𝐶𝑖 ± 𝑘. 𝐶𝑗 → 𝐶𝑖)
• The third elementary operation (no. III) does not change the
determinant of a matrix. Why?(Hint: focus on the properties of determinants)
• In order to find the inverse of a square matrix 𝑨 through the
Gaussian elimination method we attach an identity matrix 𝑰 (of the
same order) to 𝑨 and then by using a sequence of elementary row
operations on both of them matrix 𝑨 step by step transforms to an
identity matrix and the identity matrix transforms to 𝑨−𝟏
, i.e.
𝑨 ⋮ 𝑰 → 𝑰 ⋮ 𝑨−𝟏
Why?(Hint: focus on the relationship between 𝑨, 𝑰 and 𝑨−𝟏
)
Finding the Inverse of a Square Matrix
33. o Find the inverse of the matrix 𝑨 =
2 3 4
1 6 9
−1 0 1
, if it is invertible.
Applying an elementary column operation, 𝑨 can be easily calculated:
𝐶3 + 𝐶1 → 𝐶1 :
2 3 4
1 6 9
−1 0 1
→
6 3 4
10 6 9
0 0 1
; so, based on the
expansion of the last row 𝑨 = 6. Therefore, matrix 𝑨 is invertible.
To find 𝑨−𝟏, we need to make 𝑨 ⋮ 𝑰 and then follow the following
sequence of elementary row operations:
2 3 4
1 6 9
−1 0 1
1 0 0
0 1 0
0 0 1
𝑅1↔𝑅2
1 6 9
2 3 4
−1 0 1
0 1 0
1 0 0
0 0 1
−2𝑅1+𝑅2→𝑅2
𝑅1+𝑅3→𝑅3
1 6 9
0 −9 −14
0 6 10
0 1 0
1 −2 0
0 1 1
−1
9
𝑅2→𝑅2
1 6 9
0 1 14
9
0 6 10
0 1 0
−1
9
2
9 0
0 1 1
−6𝑅2+𝑅1→𝑅1
−6𝑅2+𝑅3→𝑅3
1 0 −1
3
0 1 14
9
0 0 2
3
2
3
−1
3
0
−1
9
2
9 0
2
3
−1
3
1
Finding the Inverse of a Square Matrix
34. 1 0 −1
3
0 1 14
9
0 0 2
3
2
3
−1
3
0
−1
9
2
9
0
2
3
−1
3
1
3
2
𝑅3→𝑅3
1 0 −1
3
0 1 14
9
0 0 1
2
3
−1
3
0
−1
9
2
9
0
1
−1
2
3
2
−14
9
𝑅3+𝑅2→𝑅2
1
3
𝑅3+𝑅1→𝑅1 1 0 0
0 1 0
0 0 1
1 −1
2
1
2
−5
3
1 −7
3
1 −1
2
3
2
• If the matrix 𝑨 in the above example was representing a coefficients matrix
in the system of simultaneous equations such as the following
2𝑥 + 3𝑦 + 4𝑧 = 5
𝑥 + 6𝑦 + 9𝑧 = 0
−𝑥 + 𝑧 = −4
the system could be written in the matrix form as 𝑨𝑿 = 𝑩, i.e.
2 3 4
1 6 9
−1 0 1
𝑥
𝑦
𝑧
=
5
0
−4
• And by using 𝑨−𝟏
, the unique set of solutions for the variables can be
found, because:
𝑨𝑿 = 𝑩 ⟹ 𝑨−𝟏
𝑨𝑿 = 𝑨−𝟏
𝑩 ⟹ 𝑿 = 𝑨−𝟏
𝑩
Finding the Inverse of a Square Matrix
𝑨−𝟏𝑰
35. So,
𝑥
𝑦
𝑧
=
1 −1
2
1
2
−5
3
1 −7
3
1 −1
2
3
2
5
0
−4
=
3
1
−1
→
𝑥 = 3
𝑦 = 1
𝑧 = −1
.
• The same elementary raw operations could be used to reach to the
same results:
𝑨 𝑩 → 𝑨−𝟏 𝑨 𝑨−𝟏 𝑩 → 𝑰 𝑿
c) Adjoint (Adjugate) Matrix Method:
Recall from the definition of determinant of a 3 × 3 matrix :
𝑨 = 𝑎11. 𝑪 𝟏𝟏 + 𝑎12. 𝑪 𝟏𝟐 + 𝑎13. 𝑪 𝟏𝟑
And we know that if elements in a row (column) are multiplied by non-
associated cofactors the sum of these products is zero. Using these
properties, the multiplication of square matrix 𝑨 by its transposed
cofactor matrix (called adjoint matrix, shown by adj(A))yields a scalar
matrix:
Finding the Inverse of a Square Matrix
Based on the elements of the 1st row
36. 𝑨. 𝑎𝑑𝑗 𝑨 =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
𝐶11 𝐶21 𝐶31
𝐶12 𝐶22 𝐶32
𝐶13 𝐶23 𝐶33
=
𝑨 0 0
0 𝑨 0
0 0 𝑨
= 𝑨 . 𝑰 𝟑
So,
𝑨. 𝑎𝑑𝑗 𝑨 = 𝑨 . 𝑰
or
𝑰 =
𝑨. 𝑎𝑑𝑗(𝑨)
𝑨
By multiplying both sides by 𝑨−𝟏, we have:
𝑨−𝟏 =
𝑎𝑑𝑗(𝑨)
𝑨
=
1
𝑨
. 𝑎𝑑𝑗 𝑨 =
1
𝑨
𝐶11 𝐶21 𝐶31
𝐶12 𝐶22 𝐶32
𝐶13 𝐶23 𝐶33
Finding the Inverse of a Square Matrix
37. o Find the inverse of matrix 𝑨 =
4 −1
2 −3
.
As 𝑨 = −10, the matrix is invertible. The cofactor matrix for 𝑨 can be easily
found as 𝑪 =
−3 −2
1 4
and its transposed is 𝑪′ =
−3 1
−2 4
.
So,
𝑨−𝟏 =
1
−10
−3 1
−2 4
=
0.3 −0.1
0.2 −0.4
• Clearly, the adjoint of a 2 × 2 matrix can easily be obtained by
interchanging the elements on the main diagonal (without changing the
sign) and change the sign of elements on the other diagonal (without
changing their place), i.e.
𝑩 =
𝑎 𝑏
𝑐 𝑑
→ 𝑎𝑑𝑗 𝑩 =
𝑑 −𝑏
−𝑐 𝑎
So,
𝑩−𝟏 =
𝑑
𝑩
−𝑏
𝑩
−𝑐
𝑩
𝑎
𝑩
Finding the Inverse of a Square Matrix
38. • Apart from the matrix’s inverse method, Cramer’s rule provides a
simple method of solving a simultaneous equations.
• According to this rule, the value of any variable in the system of
equation (provided that the system has a unique solution for each
variable), can be obtained through the division of two
determinants, i.e.:
𝑥 =
𝑨 𝑥
𝑨
, 𝑦 =
𝑨 𝑦
𝑨
and 𝑧 =
𝑨 𝑧
𝑨
Where 𝑨 𝑥 , 𝑨 𝑦 and 𝑨 𝑧 are specific determinants. If in 𝑨 the
column vector associated to the coefficients of any of variables is
replaced by the column vector of constants, we can obtain these
specific determinants.
Cramer’s Rule
