UNIT-4

Linear algebra

4.1 Inverse and rank of a matrix

Inverse of a matrix-

The inverse of a matrix ‘A’ can be find as-

Sol.

Inverse of a matrix by using elementary transformation-

The following transformation are defined as elementary transformations-

1. Interchange of any two rows (column)

2. Multiplication of any row or column by any non-zero scalar quantity k.

3. Addition to one row (column) of another row(column) multiplied by any non-zero scalar.

The symbol ~ is used for equivalence.

Elementary matrices-

If we get a square matrix from an identity or unit matrix by using any single elementary transformation is called elementary matrix.

Note- Every elementary row transformation of a matrix can be affected by pre multiplication with the corresponding elementary matrix.

The method of finding inverse of a non-singular matrix by using elementary transformation-

Working steps-

1. Write A = IA

2. Perform elementary row transformation of A of the left side and I on right side.

3. Apply elementary row transformation until ‘A’ (left side) reduces to I, then I reduces to

Rank of a matrix-

Rank of a matrix by echelon form-

The rank of a matrix (r) can be defined as –

1. It has at least one non-zero minor of order r.

2. Every minor of A of order higher than r is zero.

Rank of a matrix by normal form-

Any matrix ‘A’ which is non-zero can be reduced to a normal form of ‘A’ by using elementary transformations.

There are 4 types of normal forms –

The number r obtained is known as rank of matrix A.

Both row and column transformations may be used in order to find the rank of the matrix.

Note-Normal form is also known as canonical form

Sol.  We have,

Key takeaways-

4.2 Rank-nullity theorem

Let A is a matrix of order m by n, then-

Proof: If rank (A) = n, then the only solution to Ax = 0 is the trivial solution x = 0by using invertible matrix.

So that in this case null-space (A) = {0}, so nullity (A) = 0.

Now suppose rank (A) = r < n, in this case there are n – r > 0 free variable in the solution to Ax = 0.

Let represent these free variables and let denote the solution obtained by sequentially setting each free variable to 1 and the remaining free variables to zero.

Here is linearly independent.

Moreover every solution is to Ax  = 0 is a linear combination of

Which shows that spans null-space (A).

Thus is a basis for null-space(A) and nullity (A) = n – r. 

4.3 System of linear equations

There are two types of linear equations-

1.  Consistent                                   

2. Inconsistent

Let’s understand about these two types of linear equations.

Consistent –

If a system of equations has one or more than one solution, it is said be consistent.

There could be unique solution or infinite solution.

For example-

A system of linear equations-

2x + 4y = 9

x + y = 5

Has unique solution,

Whereas,

A system of linear equations-

2x + y = 6

4x + 2y = 12

Has infinite solutions.

Inconsistent-

If a system of equations has no solution, then it is called inconsistent.

Consistency of a system of linear equations-

Suppose that a system of linear equations is given as-

This is the format as AX = B

Its augmented matrix is-

[A:B] = C

(1) Consistent equations-

If Rank of A = Rank of C

Here, Rank of A = Rank of C = n ( no. of unknown) – unique solution

And Rank of A = Rank of C = r , where r<n   -  infinite solutions

(2) Inconsistent equations-

If Rank of A ≠ Rank of C

Solution of homogeneous system of linear equations-

A system of linear equations of the form AX = O is said to be homogeneous, where A denotes the coefficients and of matrix and O denotes the null vector.

Suppose the system of homogeneous linear equations is,

Note-  A system of homogeneous linear  equations always has a solution if

1. r(A) = n then there will be trivial solution, where n is the number of unknown,

2. r(A) < n , then there will be an infinite number of solution.

Solution of non-homogeneous system of linear equations-

Cramer’s rule-

This method was introduced by Gabriel Cramer (1704 – 1752).

Suppose we have to solve the following equations,

Gauss-Seidel iteration method-

Step by step method to solve the system of linear equation by using Gauss Seidal Iteration method-

Suppose,

Step-1 Here put y = 0 and z = 0  and x = in first equation

Then in second equation we put this value of x that we get the value of y.

In the third eq. we get z by using the values of x and y

Key takeaways-

A system of homogeneous linear  equations always has a solution if

1. r(A) = n then there will be trivial solution, where n is the number of unknown,

2. r(A) < n , then there will be an infinite number of solution.

4.4 Symmetric, skew-symmetric and orthogonal matrices

Symmetric matrix:

Any square matrix is said to be symmetric matrix if its transpose equals to the matrix itself.

For example:

Sol. As we know that if the transpose of the given matrix is same as the matrix itself then the matrix is called symmetric matrix.

So that, first we will find its transpose,

Example: Show that any square matrix can be expressed as the sum of symmetric matrix and anti- symmetric matrix.

Sol.  Suppose A is any square matrix.

Skew-symmetric matrix-

A square matrix A is said to be skew symmetrix matrix if –

1. A’ = -A,    [ A’ is the transpose of A]

2. All the main diagonal elements will always be zero.

This is skew symmetric matrix, because transpose of matrix A is equals to negative A

Sol. This is not a skew symmetric matrix, because the transpose of matrix A is not equals to -A.

-A = A’

Orthogonal matrix-

Any square matrix A is said to be an orthogonal matrix if the product of the matrix A and its transpose is an identity matrix.

Such that,

A. A’ = I

Matrix × transpose of matrix = identity matrix

Note- if |A| = 1, then we can say that matrix A is proper.

4.5 Determinants

Look at the following procedure in order to understand the concept of determinants-

Let’s consider the two linear equations,

Now suppose we have three equations as below, where a, y and z are unknown,

Minor –

The minor of an element is define as determinant obtained by deleting the row and column containing the element.

In a determinant,

Cofactor –

Cofactors can be defined as follows,

                          Cofactor = minor

Where r is the number of rows and c is the number of columns.

Sol.  (1)  The minor of element 2 will be,

Delete the corresponding row and column of element 2,

Properties of determinants-

(1) If the rows are interchanged into columns or columns into rows then the value of determinants does not change.

Let us consider the following determinant:

(2) The sign of the value of determinant changes when two rows or two columns are interchanged.

Interchange the first two rows of the following, we get

(3) If two rows or two columns are identical the the value of determinant will be zero.

Let, the determinant has first two identical rows,

As we know that if we interchange the first two rows then the sign of the value of the determinant will be changed , so that

 (4) if the element of any row of a determinant be each multiplied by the same number then the determinant multiplied by the same number,

Example: prove the following without expanding the determinant.

Example: prove that (without expanding) the determinant given below is equal zero.

Applications of determinants-

Determinants have various applications such as finding the area and condition of collinearity.

Area of triangles-

Suppose the three vertices of a triangle are respectively, then we know that the area of the triangle is given by-

This is how we can find the area of the triangle.

Condition of collinearity-

Let there are three points

Then these three points will be collinear if -

Example: Show that the points given below are collinear-

Sol.

First we need to find the area of these points and if the area is zero then we can say that these are collinear points-

So that-

We know that area enclosed by three points-

So that these points are collinear.

Key takeaways-

4.6 Eigenvalues and eigenvectors

First we will go through some important definitions before studying Eigen values and Eigen vectors.

1. Vector-

An ordered n – touple of numbers is called an n – vector. Thus the ‘n’ numbers x1, x2, …………xn taken in order denote the vector x. i.e. x = (x1, x2, ……., xn).

Where the numbers x1, x2, ………..,xn are called component or co – ordinates of a vector x. A vector may be written as row vector or a column vector.

If A be an mxn matrix then each row will be an n – vector & each column will be an m – vector.

2. Linear dependence-

A set of n – vectors. x1, x2, …….., xr is said to be linearly dependent if there exist scalars. k1, k2, …….,kr not all zero such that

k1 + x2k2 + …………….. + xrkr = 0     … (1)

3. Linear independence-

A set of r vectors x1, x2, ………….,xr is said to be linearly independent if there exist scalars k1, k2, …………, kr all zero such that

x1 k1 + x2 k2 + …….. + xrkr = 0

Important notes-

1. Equation (1) is known as a vector equation.
2. If the vector equation has non – zero solution i.e. k1, k2, ….,kr not all zero. Then the vector x1, x2, ……….xr are said to be linearly dependent.
3. If the vector equation has only trivial solution i.e.

k1 = k2 = …….=kr = 0. Then the vector x1, x2, ……,xr are said to linearly independent.

4. Linear combination-

A vector x can be written in the form.

x = x1 k1 + x2 k2 + ……….+xrkr

where k1, k2, ………….., kr are scalars, then X is called linear combination of x1, x2, ……, xr.

Results:

1. A set of two or more vectors are said to be linearly dependent if at least one vector can be written as a linear combination of the other vectors.
2. A set of two or more vector are said to be linearly independent then no vector can be expressed as linear combination of the other vectors.

Example 1

Are the vectors , , linearly dependent. If so, express x1 as a linear combination of the others.

Solution:

Example 2

Examine whether the following vectors are linearly independent or not.

and .

Solution:

Consider the vector equation,

Example 3

At what value of P the following vectors are linearly independent.

Solution:

Note:-

If the rank of the coefficient matrix is r, it contains r linearly independent variables & the remaining vectors (if any) can be expressed as linear combination of these vectors.

Characteristic equation:-

Let A he a square matrix, be any scalar then is called characteristic equation of a matrix A.

Note:

Let a be a square matrix and ‘’ be any scalar then,

The roots of a characteristic equations are known as characteristic root or latent roots, Eigen values or proper values of a matrix A.

Eigen vector:-

Suppose be an Eigen value of a matrix A. Then a non – zero vector x1 such that.

Such a vector ‘x1’ is called as Eigen vector corresponding to the Eigen value .

Properties of Eigen values:-

1. Then sum of the Eigen values of a matrix A is equal to sum of the diagonal elements of a matrix A.
2. The product of all Eigen values of a matrix A is equal to the value of the determinant.
3. If are n Eigen values of square matrix A then are m Eigen values of a matrix A-1.
4. The Eigen values of a symmetric matrix are all real.
5. If all Eigen values are non –zero then A-1 exist and conversely.
6. The Eigen values of A and A’ are same.

Properties of Eigen vector:-

1. Eigen vector corresponding to distinct Eigen values are linearly independent.
2. If two are more Eigen values are identical then the corresponding Eigen vectors may or may not be linearly independent.
3. The Eigen vectors corresponding to distinct Eigen values of a real symmetric matrix are orthogonal.

Example-1:

4.7 Diagonalization of matrices

Two square matrix and A of same order n are said to be similar if and only if

for some non singular matrix P.

Such transformation of the matrix A into with the help of non singular matrix P is known as similarity transformation.

Similar matrices have the same Eigen values.

If X is an Eigen vector of matrix A then is Eigen vector of the matrix

Reduction to Diagonal Form:

Let A be a square matrix of order n has n linearly independent Eigen vectors Which form the matrix P such that

Where P is called the modal matrix and D is known as spectral matrix.

Procedure: let A be a square   matrix of order 3.

Let three Eigen vectors of A

Corresponding to Eigen values

Note: The method of diagonalization is helpful in calculating power of a matrix.

  .Then for an integer n we have  

We are using the example of 1.6*

Key takeaways-

1. Two square matrix and A of same order n are said to be similar if and only if

for some non singular matrix P.

2.     Let A be a square matrix of order n has n linearly independent Eigen vectors Which form the matrix P such that

Where P is called the modal matrix and D is known as spectral matrix.

4.8 Cayley-Hamilton Theorem, and Orthogonal transformation

Statement-

Every square matrix satisfies its characteristic equation , that means for every square matrix of order n,

Sol. Characteristic equation of the matrix, we can be find as follows-

Inverse of a matrix by Cayley-Hamilton theorem-

We can find the inverse of any matrix by multiplying the characteristic equation With .

For example, suppose we have a characteristic equation   then multiply this by  , then it becomes

Then we can find by solving the above equation.

Orthogonal transformation-

When a transformation from one set of rectangular coordinates to another set of rectangular coordinates  , then this is called ‘orthogonal tranformation’.

In other words-

A linear transformation Y = AX is said to be orthogonal , if matrix A is orthogonal that means,

If Y = AX is an orthogonal transformation , then we have

        X’X = Y’Y = (AX)’AX = X’A’ AX = X’(A’A)X

Which is only possible , if

A’A = I = AA’  and

Therefore the square matrix A is said to be orthogonal if AA’ = A’A and

NOTES-

1. The inverse and transpose of an orthogonal matrix is also orthogonal.

2. A linear transformation preserves length if and only if its matris is orthogonal.

3. The product of two or more than two orthogonal matrices is also orthogonal.

4. The determinant of an orthogonal matrix will always be ±1.

Key takeaways-

1. 
2. Orthogonal transformation-

When a transformation from one set of rectangular coordinates to another set of rectangular coordinates  , then this is called ‘orthogonal tranformation’.

References

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