The document discusses key concepts in linear algebra, including the rank of a matrix, which indicates the maximum number of linearly independent rows or columns, and is important for solving linear systems. It also covers the Cayley-Hamilton theorem, which asserts that any square matrix satisfies its own characteristic polynomial, resulting in the zero matrix. Additionally, the document explains diagonalization of a matrix, which involves finding a matrix that transforms a square matrix into a diagonal form if it has enough linearly independent eigenvectors.

1. content
 RANK OF MATRIX
 CAYLEY-HAMILTON THEOREM
 DIAGONALIZATION OF MATRIX
Matrices
Unit 1
Participants
Mayank Hargude IT58
Mayank Patil IT59
Mohammad Aqdas IT61
Mayur Sawarbandhe IT60

2.  RANK OF MATRIX
 The rank of a matrix is the maximum
number of linearly independent rows or
columns in the matrix. In simpler terms,
it represents the dimension of the space
spanned by the rows or columns. A
matrix's rank is essential in various
mathematical applications, such as
solving linear systems of equations and
understanding the properties of
transformations.
 For example, in a 3x3 matrix, if two rows
are linearly dependent, the rank would
be 2. The same applies to columns. The
rank provides insights into the matrix's
structure and can be determined through
various methods, such as row reduction
or using the determinant of minors.

3. CAYLEY
-HAMIL
TON THEOREM
The Cayley-Hamilton theorem states that for any square matrix A,
substituting the matrix into its own characteristic polynomial yields
the zero matrix. In other words, p(A) = 0, where p(λ) is the
characteristic polynomial of (A).

5. DIAGONALIZATION OF MATRIX
BAB =
1
If ‘A
’ is square matrix of order n having n linearly independent Eigen
vector, then a non singular matrix B can be found such that B-1
AB is a
diagonal form . If A is a square of order 3, having 3 linearly independent
Eigen Vectors corresponding to Eigen Value λ1, λ2 , λ3 then
[λ 0 0
]
0 λ 0
0 0 λ
Where B is Eigen Vector Matrix for Eigen value λ1 , λ2, λ3 and B is called
Model Matrix .

7. Thank
you