2. Linear Transformation
โข Matrix Algebra developed in relation to linear transformations such
as the following:
๐๐ฅ + ๐๐ฆ = ๐
๐๐ฅ + ๐๐ฆ = ๐
Where ๐, ๐, ๐ and ๐ are real numbers. This transformation introduces
a function(mapping) by which an ordered pair ๐ฅ, ๐ฆ in ๐ฅ๐๐ฆ plane
transformed (associated) to another ordered pair (๐, ๐) in ๐๐๐ plane.
๐ ๐ฟ
๐๐
๐ ๐ถ
This linear transformation can
be done through the coefficients
of ๐ฅ and ๐ฆ . The square array
๐ ๐
๐ ๐
represents this
transformation which is one
among many other
transformations. Such an array
is called matrix.
3. Matrix Algebra
โข Definition: A matrix is a rectangular or square array of
elements (usually numbers) arranged in rows and
columns.
โข 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 ๐จ ๐ร๐ and matrix B with m rows and n columns
is shown by ๐ฉ ๐ร๐. Their elements are shown by small
letters with an index indicating the position of the
element in the matrix.
โข ๐ด3ร2 =
๐11
๐21
๐31
๐12
๐22
๐32
๐ต ๐ร๐ =
๐11
๐21
โฎ
๐ ๐1
๐12
๐22
โฎ
๐ ๐2
๐13 โฆ
๐23 โฆ
โฎ โฏ
๐ ๐3 โฏ
๐1๐
๐2๐
โฎ
๐ ๐๐
4. Matrix Algebra
โข There are other ways of showing a matrix:
๐ฉ = ๐๐๐ ๐ร๐
๐๐ ๐ฉ ๐ร๐
The Order of a Matrix:
โข 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.
โข In the previous examples the order of A is 3 ร 2 and the
order of B is ๐ ร ๐.
โข If ๐ = ๐ then the matrix is called a square matrix of order
๐ (๐๐ ๐).
5. Vectors & Scalars
โข A matrix with just one row or one column is called vector.
๐ด1ร3 = 2 โ10 3.5 is a row (horizontal) vector.
๐ต4ร1 =
2
โ1.65
7.2
5
is a column (vertical) vector.
โข In matrix algebra any real number is called scalar. So, a
scalar in matrix algebra is a 1 ร 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 ๐ .
๐ด2ร2 =
0 0
0 0
๐ถ2ร3 =
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.
๐ด3ร3 =
3 0 0
0 โ1 0
0 0 2
Main Diagonal๐ = ๐ โ ๐๐๐ โ ๐
๐ โ ๐ โ ๐๐๐ = ๐
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.
๐ผ2ร2 = ๐ผ2 =
1 0
0 1
๐ผ3ร3 = ๐ผ3 =
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.
๐ด3ร3 =
3 0 0
0 3 0
0 0 3
8. Types of Matrices
Transpose Matrix:
For a matrix ๐จ ๐ร๐ the transpose is defined as ๐จโฒ ๐ร๐ (in some books ๐จ ๐ร๐
๐ป
)
where the rows and columns are interchanged.
๐ด2ร4 =
1 4
3 โ2
1
0
โ3
1.2
โ ๐ดโฒ4ร2 =
1
4
3
โ2
1
โ3
0
1.2
โข Transposed of a row vector is a column vector and vice versa.
๐3ร1 =
1
5
4
โ ๐โฒ1ร3 = 1 5 4
Properties of Transpose Matrix:
๏ง By the definition of transpose matrix we can conclude ๐จโฒ โฒ = ๐จ.
๏ง By the definition, ๐ฐโฒ = ๐ฐ. This property is true for all diagonal matrices.
๏ง For a square matrix ๐จ, if ๐จโฒ
= ๐จ , then ๐จ is a symmetric matrix.
1 0.5
0.5 3
๏ง ๐๐จ โฒ = ๐๐จโฒ
9. Types of Matrices
Triangular Matrices:
If all elements above the main diagonal of a square matrix are
zero the matrix is called โlower triangular matrixโ.
e.g. ๐ด =
2 0 0
0 โ1 0
4 3 5
if ๐ < ๐ , ๐๐๐ = 0
Alternatively, If all elements under the main diagonal of a
square matrix are zero the matrix is called โupper triangular
matrixโ.
e.g. ๐ต =
1 โ3 1
2
0 4 7
0 0 โ6
if ๐ > ๐ , ๐๐๐ = 0
10. Types of Matrices
Symmetric Matrix:
A square matrix is symmetric if ๐จ = ๐จโฒ. 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)
๐ด3ร3 =
3 1.2 2
1.2 โ1 0
2 0 2
Equality in matrices:
โข Two matrices ๐จ and ๐ฉ are equal if they have the same order
and their corresponding elements are equal.
๐จ = ๐ฉ โ
๐๐๐๐๐ ๐จ = ๐๐๐๐๐(๐ฉ)
โ๐, ๐ โ ๐๐๐ = ๐๐๐
11. Matrix Operation
Scalar Multiplication:
If ๐ is a scalar then
๐. ๐จ = ๐. ๐๐๐ ๐ร๐
This means that all elements of the matrix are multiplied by
the scalar ๐.
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:
๐จ ๐ร๐ ยฑ ๐ฉ ๐ร๐ = ๐๐๐ ยฑ ๐๐๐ ๐ร๐
12. Matrix Operations
e.g. ๐ด =
3 1 โ2
2 4 1
and ๐ต =
7 โ10 4
5 0 3
๐ด + ๐ต =
10 โ9 2
7 4 4
๐ด โ ๐ต =
โ4 11 โ6
โ3 4 โ2
Properties of Addition & Subtraction:
๏ง ๐จ + ๐ฉ = ๐ฉ + ๐จ Commutative law
๏ง ๐จ ยฑ ๐ฉ ยฑ ๐ช = ๐จ ยฑ ๐ฉ ยฑ ๐ช Associative law
๏ง ๐. ๐จ ยฑ ๐ฉ = ๐๐จ ยฑ ๐๐ฉ (๐ is a scalar)
๏ง ๐จ ยฑ ๐ฉ โฒ = ๐จโฒ ยฑ ๐ฉโฒ can be extended to โnโ
matrices
13. Matrix Operations
โข Matrix Multiplication:
Multiplication of two matrices ๐จ and ๐ฉ, in the form of ๐จ ร ๐ฉ or ๐จ๐ฉ, is
possible if the number of columns in ๐จ is equal to the number of rows
in ๐ฉ. The result of this multiplication is another matrix ๐ช where the
number of its rows is equal to the number of rows in ๐จ and number of
its columns is equal to the number of columns in ๐ฉ; that is:
๐จ ๐ร๐ ร ๐ฉ ๐ร๐ = ๐ช ๐ร๐
Elements of ๐ช can be calculated by adding some multiplications;
multiplications of the elements in the i-th row of ๐จ by the
corresponding elements in the j-th column of ๐ฉ, that is:
๐ช๐๐ = ๐=1
๐
๐๐๐ ๐ ๐๐ where
๐ = 1,2, โฏ , ๐
๐ = 1,2, โฏ , ๐
14. Matrix Operations
โข For example, matrix ๐จ ๐ร๐ =
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
cannot be multiplied by a
horizontal vector ๐ฟ ๐ร๐ = ๐ฅ ๐ฆ ๐ง but it can be multiplied by its
transpose which is a vertical vector; ๐ฟโฒ ๐ร๐ =
๐ฅ
๐ฆ
๐ง
and the result is:
AX =
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
๐ฅ
๐ฆ
๐ง
=
๐๐ฅ + ๐๐ฆ + ๐๐ง
๐๐ฅ + ๐๐ฆ + ๐๐ง
๐๐ฅ + โ๐ฆ + ๐๐ง
โข In the above example:
๐ฟ๐ฟโฒ
= ๐ฅ2
+ ๐ฆ2
+ ๐ง2
which is a scalar but ๐ฟโฒ
๐ฟ =
๐ฅ2 ๐ฅ๐ฆ ๐ฅ๐ง
๐ฆ๐ฅ ๐ฆ2 ๐ฆ๐ง
๐ง๐ฅ ๐ง๐ฆ ๐ง2
which is a symmetric matrix, why?
15. Matrix Operations
Properties of Matrix Multiplication:
๏ง In general, ๐จ๐ฉ โ ๐ฉ๐จ if both exist, but there are special cases that
this property is not true.
๏ง If ๐ฐ is an identity matrix ๐ฐ๐ฉ = ๐ฉ๐ฐ = ๐ฉ.
๏ง ๐จ ๐ฉ + ๐ช = ๐จ๐ฉ + ๐จ๐ช and ๐ฉ + ๐ช ๐จ = ๐ฉ๐จ + ๐ช๐จ
๏ง ๐จ ๐ฉ๐ช = ๐จ๐ฉ ๐ช
๏ง If ๐จ๐ฉ exist then ๐จ๐ฉ โฒ = ๐ฉโฒ ๐จโฒ (this can be extended to more than 2
matrices, i.e.: ๐จ๐ฉ๐ช โฒ
= ๐ชโฒ๐ฉโฒ
๐จโฒ
๏ง From ๐จ๐ฉ = ๐ we cannot conclude necessarily that ๐จ = ๐๐๐ ๐ฉ = ๐.*
๏ง From ๐จ๐ฉ = ๐จ๐ช we cannot conclude necessarily that ๐ฉ = ๐ช.**
16. Determinant of a Matrix
โข Consider the system of simultaneous equations
๐๐ + ๐๐ = ๐
๐๐ + ๐ ๐ = ๐
Where ๐, ๐, โฆ . , ๐, ๐ are constants of the system. If the coefficients of
๐ and ๐ in the first equation (i.e. ๐ and ๐ )have a linear relationship
with the coefficients of the second equation (i.e. ๐ and ๐ ), the system
either does not have a unique solutions for ๐ and ๐ (when ๐, ๐ 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).
โข If
๐
๐
=
๐
๐
โ ๐๐ = ๐๐ or ๐๐ โ ๐๐ = 0 it means the
coefficients have a linear relationship and there is no
unique solutions for ๐ฅ and ๐ฆ. The value of ๐๐ โ ๐๐
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:
2๐ฅ + 3๐ฆ = 12
4๐ฅ + 6๐ฆ = 24
and
B:
2๐ฅ + 3๐ฆ = 12
4๐ฅ + 6๐ฆ = โ18
we have:
2
4
=
3
6
โ 2 ร 6 = 3 ร 4 ๐๐ 2 ร 6 โ 3 ร 4 = 0
So, both systems fail to provide unique
solutions for ๐ฅ and ๐ฆ 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
2๐ฅ + 3๐ฆ = 12
4๐ฅ + 6๐ฆ = 24
2๐ฅ + 3๐ฆ = 12
4๐ฅ + 6๐ฆ = โ18
x
y
Infinite solutions
No solution
18. Determinant of a Matrix
โข for matrix ๐จ ๐ร๐ =
๐ ๐
๐ ๐
, the value of ๐๐ โ ๐๐ is called
โdeterminantโ of the matrix and it is shown by det ๐จ or simply ๐จ .
๐จ ๐ร๐=
๐ ๐
๐ ๐
โ det ๐จ = ๐จ = ๐๐ โ ๐๐
โข 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.
โข What about if the square matrix is ๐ ร ๐ or even ๐ ร ๐?
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
โข Minors: For every element (such as ๐๐๐) of a square matrix there
is a corresponding determinant, called โminor of ๐๐๐โ (shown by
๐๐๐) derived from ignoring the elements in the same row and
column of ๐๐๐ (i.e. ๐ and ๐).
โข For matrix
๐11 ๐12 ๐13
๐21 ๐22 ๐23
๐31 ๐32 ๐33
, minors are:
Minor of ๐11 = ๐11 =
๐22 ๐23
๐32 ๐33
= ๐22 ๐33 โ ๐23 ๐32
Minor of ๐12 = ๐12 =
๐21 ๐23
๐31 ๐33
= ๐21 ๐33 โ ๐23 ๐31
Minor of ๐13 = ๐13 =
๐21 ๐22
๐31 ๐32
= ๐21 ๐32 โ ๐22 ๐31
Minor of ๐21 = ๐21 =
๐12 ๐13
๐32 ๐33
= ๐12 ๐33 โ ๐13 ๐32
20. Determinant of Matrices of Higher Orders than 2
โข Minor of ๐22 = ๐22 =
๐11 ๐13
๐31 ๐33
= ๐11 ๐33 โ ๐13 ๐31
โข โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ
โข โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ
โข Minor of ๐33 = ๐33 =
๐11 ๐12
๐21 ๐22
= ๐11 ๐22 โ ๐12 ๐21
โข Cofactors: Cofactors of each element ๐๐๐, shown by ๐ถ๐๐, are minors with a
sign depending on the row and column of the element. i.e.:
๐ถ๐๐ = โ1 ๐+๐ ๐๐๐
So,
the cofactor of ๐11 is ๐ช ๐๐ = โ1 1+1
๐11 = ๐11 = ๐22 ๐33 โ ๐23 ๐32
And
the cofactor of ๐23 is
๐ช ๐๐ = โ1 2+3 ๐23 = โ๐23= โ(๐11 ๐32 โ ๐12 ๐31) = โ๐11 ๐32 + ๐12 ๐31
21. Determinant of Matrices of Higher Orders than 2
โข The matrix of cofactors can be shown as:
๐ถ =
๐ถ11 ๐ถ12 ๐ถ13
๐ถ21 ๐ถ22 ๐ถ23
๐ถ31 ๐ถ32 ๐ถ33
=
๐11 โ๐12 ๐13
โ๐21 ๐22 โ๐23
๐31 โ๐32 ๐33
Now, we can define and calculate the determinant of a matrix with
order higher than two.
Definition: Determinant of a ๐ ร ๐ matrix is the summation of
products between elements of any row (or any column ) and their
corresponding cofactors. i.e.:
For a matrix ๐จ ๐ร๐ we can write:
๐จ = ๐11. ๐ช ๐๐ + ๐12. ๐ช ๐๐ + โฏ + ๐1๐ . ๐ช ๐๐ Based on the 1st row
๐จ = ๐1๐. ๐ช ๐๐ + ๐2๐. ๐ช ๐๐ + โฏ + ๐ ๐๐ . ๐ช ๐๐ Based on the nth column
22. Determinant of Matrices of Higher Orders than 2
o Find the determinant of ๐ =
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
.
Based on the elimination of rows and columns using the elements
of the first row we have:
๐จ = ๐.
๐ ๐
โ ๐
โ ๐.
๐ ๐
๐ ๐
+ ๐.
๐ ๐
๐ โ
= ๐ ๐๐ โ ๐โ โ ๐ ๐๐ โ ๐๐ + ๐(๐โ โ ๐๐)
= ๐๐๐ โ ๐๐โ โ ๐๐๐ + ๐๐๐ + ๐๐โ โ ๐๐๐
o The determinant of the unit matrix of order ๐ is:
๐ฐ ๐ร๐ = ๐ฐ ๐ =
1
0
0 โฆ
1 โฏ
0
0
โฎ โฎ โฏ โฎ
0 0 โฆ 1
๐ฐ ๐ = ๐ฐ ๐โ๐ = โฏ = ๐ฐ ๐ = 1 , why?
23. Sarrusโ Rule
โข For a matrix ๐ =
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
can be calculated through following steps:
1. Add the first 2 columns of the matrix to the right of the 3rd column:
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
๐
๐
๐
๐
๐
โ
2. Subtract the sum of the products along the green arrows from the sum of
products along the blue arrows:
๐จ = ๐๐๐ + ๐๐๐ + ๐๐โ โ (๐๐๐ + ๐๐โ + ๐๐๐)
โข Note: It is also possible to add the first 2 rows of the matrix to the bottom of
the 3rd row:
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
๐ ๐ ๐
๐ ๐ ๐
(+) (-)
(+) (-)
๐จ = ๐๐๐ + ๐๐๐ + ๐๐โ โ (๐๐๐ + ๐๐โ + ๐๐๐)
24. Properties of Determinants
1) Transposing a matrix does not change its determinant: ๐จ = ๐จโฒ
๐ ๐
๐ ๐
=
๐ ๐
๐ ๐
= ๐๐ โ ๐๐
2) If all elements of a row (or column) of a square matrix are zero the
determinant of that matrix is zero. Why?
๐ 0 2
๐ 0 3
๐ 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.
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
=
๐ ๐ ๐
๐๐ ๐๐ ๐๐
๐ โ ๐
= 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 ๐ =
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
then
๐. ๐ ๐ ๐
๐. ๐ ๐ ๐
๐. ๐ โ ๐
= ๐. ๐จ and
๐. ๐ ๐. ๐ ๐. ๐
๐. ๐ ๐. ๐ ๐. ๐
๐. ๐ ๐. โ ๐. ๐
=
๐3.
๐ ๐ ๐
๐ ๐ ๐
๐ โ ๐
๐๐ ๐. ๐จ = ๐3. ๐จ
If matrix ๐จ was from
order of ๐ then
๐. ๐จ = ๐ ๐. ๐จ
26. Properties of Determinants
5) For the square matrices ๐จ and ๐ฉ with the same orders
๐จ๐ฉ = ๐จ . ๐ฉ
6) If two rows (or two columns) of a square matrix are interchanged
the determinant of the matrix is multiplied by -1.
๐ ๐
๐ ๐
= โ
๐ ๐
๐ ๐
๐๐๐ก๐๐๐โ๐๐๐๐๐๐ ๐ก๐ค๐ ๐๐๐ค๐
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.:
๐ + ๐ ๐ + ๐
๐ ๐
=
๐ ๐
๐ ๐
+
๐ ๐
๐ ๐
๐ + ๐ ๐
๐ + ๐ ๐
=
๐ ๐
๐ ๐
+
๐ ๐
๐ ๐
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.
๐ + ๐. ๐ ๐
๐ + ๐. ๐ ๐
=
๐ ๐
๐ ๐
+ ๐.
๐ ๐
๐ ๐
=
0
๐ ๐
๐ ๐
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 โ2 5
0 0 3
= 1 ร โ2 ร 3 = โ6
Diagonal matrix:
1 0 0
0 โ2 0
0 0 3
= 1 ร โ2 ร 3 = โ6
Scalar Matrix:
โ2 0 0
0 โ2 0
0 0 โ2
= โ2 ร ๐ผ3 = โ2 3 ร ๐ฐ ๐
1
= โ8
Properties of Determinants
28. โข The last two properties are sometimes used to facilitate the
calculation of determinant of a matrix.
o If ๐จ =
2 3 โ1
1 4 0
โ3 5 4
find ๐จ .
According to the property No. 8, if we substitute the last row (๐ 3) by
4๐ 1 + ๐ 3 (multiplying the first row by 4 and adding it to the third
row) the result of the determinant does not change. So:
2 3 โ1
1 4 0
โ3 5 4
=
2 3 โ1
1 4 0
5 17 0
= โ1 ร
1 4
5 17
= 3
โข 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. โข 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.
โข In matrix algebra the inverse of a square matrix ๐จ, which is shown by
๐จโ๐
(read ๐จ inverse), is the matrix of the same order such that:
๐จ๐จโ๐
= ๐จโ๐
๐จ = ๐ฐ
Where ๐ฐ 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:
๏ง ๐จโ๐ โ๐
= ๐จ
๏ง ๐จ๐ฉ โ๐
= ๐ฉโ๐
๐จโ๐
๏ง ๐จโฒ โ๐
= ๐จโ๐ โฒ
๏ง ๐จ๐จโ๐
= ๐ฐ โ ๐จ . ๐จโ๐
= 1 โ ๐จโ๐
=
1
๐จ
Invers of a Matrix
30. ๏ง A square matrix ๐จ is invertible if and only if ๐จ โ 0. This is
necessary and sufficient condition for a square matrix to have an
inverse. If ๐จ โ 0, the matrix is called non-singular and singular
otherwise.
๏ง 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 ๐จ =
2 4
5 5
.
As ๐จ = โ10, so, the inverse exists. According to the definition, if
๐จโ๐
=
๐ ๐
๐ ๐
then : ๐๐จโ๐
=
2 4
5 5
๐ ๐
๐ ๐
=
1 0
0 1
= ๐ฐ. By
multiplication we have:
2๐ + 4๐ 2๐ + 4๐
5๐ + 5๐ 5๐ + 5๐
=
1 0
0 1
By solving the system of four simultaneous equations with four
variables we will have : ๐ = โ0.5 , ๐ = โ0.5 , ๐ = 0.5 and ๐ = โ0.5.
Finding the Inverse of a Square Matrix
31. So, ๐จโ๐ =
โ0.5 โ0.5
0.5 โ0.5
. 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. ๐ ๐ โ ๐ ๐ (๐ถ๐ โ ๐ถ๐)
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.
๐. ๐ ๐ โ ๐ ๐ (๐. ๐ถ๐ โ ๐ถ๐)
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. ๐ ๐ ยฑ ๐. ๐ ๐ โ ๐ ๐ (๐ถ๐ ยฑ ๐. ๐ถ๐ โ ๐ถ๐)
โข The third elementary operation (no. III) does not change the
determinant of a matrix. Why?(Hint: focus on the properties of determinants)
โข In order to find the inverse of a square matrix ๐จ through the
Gaussian elimination method we attach an identity matrix ๐ฐ (of the
same order) to ๐จ and then by using a sequence of elementary row
operations on both of them matrix ๐จ step by step transforms to an
identity matrix and the identity matrix transforms to ๐จโ๐
, i.e.
๐จ โฎ ๐ฐ โ ๐ฐ โฎ ๐จโ๐
Why?(Hint: focus on the relationship between ๐จ, ๐ฐ and ๐จโ๐
)
Finding the Inverse of a Square Matrix
33. o Find the inverse of the matrix ๐จ =
2 3 4
1 6 9
โ1 0 1
, if it is invertible.
Applying an elementary column operation, ๐จ can be easily calculated:
๐ถ3 + ๐ถ1 โ ๐ถ1 :
2 3 4
1 6 9
โ1 0 1
โ
6 3 4
10 6 9
0 0 1
; so, based on the
expansion of the last row ๐จ = 6. Therefore, matrix ๐จ is invertible.
To find ๐จโ๐, we need to make ๐จ โฎ ๐ฐ and then follow the following
sequence of elementary row operations:
2 3 4
1 6 9
โ1 0 1
1 0 0
0 1 0
0 0 1
๐ 1โ๐ 2
1 6 9
2 3 4
โ1 0 1
0 1 0
1 0 0
0 0 1
โ2๐ 1+๐ 2โ๐ 2
๐ 1+๐ 3โ๐ 3
1 6 9
0 โ9 โ14
0 6 10
0 1 0
1 โ2 0
0 1 1
โ1
9
๐ 2โ๐ 2
1 6 9
0 1 14
9
0 6 10
0 1 0
โ1
9
2
9 0
0 1 1
โ6๐ 2+๐ 1โ๐ 1
โ6๐ 2+๐ 3โ๐ 3
1 0 โ1
3
0 1 14
9
0 0 2
3
2
3
โ1
3
0
โ1
9
2
9 0
2
3
โ1
3
1
Finding the Inverse of a Square Matrix
34. 1 0 โ1
3
0 1 14
9
0 0 2
3
2
3
โ1
3
0
โ1
9
2
9
0
2
3
โ1
3
1
3
2
๐ 3โ๐ 3
1 0 โ1
3
0 1 14
9
0 0 1
2
3
โ1
3
0
โ1
9
2
9
0
1
โ1
2
3
2
โ14
9
๐ 3+๐ 2โ๐ 2
1
3
๐ 3+๐ 1โ๐ 1 1 0 0
0 1 0
0 0 1
1 โ1
2
1
2
โ5
3
1 โ7
3
1 โ1
2
3
2
โข If the matrix ๐จ in the above example was representing a coefficients matrix
in the system of simultaneous equations such as the following
2๐ฅ + 3๐ฆ + 4๐ง = 5
๐ฅ + 6๐ฆ + 9๐ง = 0
โ๐ฅ + ๐ง = โ4
the system could be written in the matrix form as ๐จ๐ฟ = ๐ฉ, i.e.
2 3 4
1 6 9
โ1 0 1
๐ฅ
๐ฆ
๐ง
=
5
0
โ4
โข And by using ๐จโ๐
, the unique set of solutions for the variables can be
found, because:
๐จ๐ฟ = ๐ฉ โน ๐จโ๐
๐จ๐ฟ = ๐จโ๐
๐ฉ โน ๐ฟ = ๐จโ๐
๐ฉ
Finding the Inverse of a Square Matrix
๐จโ๐๐ฐ
35. So,
๐ฅ
๐ฆ
๐ง
=
1 โ1
2
1
2
โ5
3
1 โ7
3
1 โ1
2
3
2
5
0
โ4
=
3
1
โ1
โ
๐ฅ = 3
๐ฆ = 1
๐ง = โ1
.
โข The same elementary raw operations could be used to reach to the
same results:
๐จ ๐ฉ โ ๐จโ๐ ๐จ ๐จโ๐ ๐ฉ โ ๐ฐ ๐ฟ
c) Adjoint (Adjugate) Matrix Method:
Recall from the definition of determinant of a 3 ร 3 matrix :
๐จ = ๐11. ๐ช ๐๐ + ๐12. ๐ช ๐๐ + ๐13. ๐ช ๐๐
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 ๐จ 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. ๐จ. ๐๐๐ ๐จ =
๐11 ๐12 ๐13
๐21 ๐22 ๐23
๐31 ๐32 ๐33
๐ถ11 ๐ถ21 ๐ถ31
๐ถ12 ๐ถ22 ๐ถ32
๐ถ13 ๐ถ23 ๐ถ33
=
๐จ 0 0
0 ๐จ 0
0 0 ๐จ
= ๐จ . ๐ฐ ๐
So,
๐จ. ๐๐๐ ๐จ = ๐จ . ๐ฐ
or
๐ฐ =
๐จ. ๐๐๐(๐จ)
๐จ
By multiplying both sides by ๐จโ๐, we have:
๐จโ๐ =
๐๐๐(๐จ)
๐จ
=
1
๐จ
. ๐๐๐ ๐จ =
1
๐จ
๐ถ11 ๐ถ21 ๐ถ31
๐ถ12 ๐ถ22 ๐ถ32
๐ถ13 ๐ถ23 ๐ถ33
Finding the Inverse of a Square Matrix
37. o Find the inverse of matrix ๐จ =
4 โ1
2 โ3
.
As ๐จ = โ10, the matrix is invertible. The cofactor matrix for ๐จ can be easily
found as ๐ช =
โ3 โ2
1 4
and its transposed is ๐ชโฒ =
โ3 1
โ2 4
.
So,
๐จโ๐ =
1
โ10
โ3 1
โ2 4
=
0.3 โ0.1
0.2 โ0.4
โข Clearly, the adjoint of a 2 ร 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.
๐ฉ =
๐ ๐
๐ ๐
โ ๐๐๐ ๐ฉ =
๐ โ๐
โ๐ ๐
So,
๐ฉโ๐ =
๐
๐ฉ
โ๐
๐ฉ
โ๐
๐ฉ
๐
๐ฉ
Finding the Inverse of a Square Matrix
38. โข Apart from the matrixโs inverse method, Cramerโs rule provides a
simple method of solving a simultaneous equations.
โข 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.:
๐ฅ =
๐จ ๐ฅ
๐จ
, ๐ฆ =
๐จ ๐ฆ
๐จ
and ๐ง =
๐จ ๐ง
๐จ
Where ๐จ ๐ฅ , ๐จ ๐ฆ and ๐จ ๐ง are specific determinants. If in ๐จ 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.
Cramerโs Rule