matrix class 12th notes for revision

matrix class 12th notes for revision 

•  matrix is defined as the arrangement in which some numbers and some letters or any other things are arranged in form of rows and columns.
• A is arranged as in orders and columns.
• A is defined as a11, a12, a13
•                          a21 , a22, a23
•                          a31 , a32 , a33
• 
  order is always written as = order (multiple )* column ☝.
  
• types of the matrix are of various types:-:-\
• 
  row matrix:- any matrix having a single row is known as row matrix.
  
  
  
• column matrix:- any matrix having a single column is known as a column matrix.
• zero matrices: this type of matrix is also called empty matrix and null matrix, represented by 0, any matrix having all the elements are 0 . that means each and every element would be zero in this type of matrix. 
• square matrix:- any matrix having the same number of rows and columns is known as a square matrix. 
• the elements of the matrix or the elements inside the matrix are called Aij, I into j  elements where I stands for rows and j stands for the column.
• diagonal matrix:- this is a type of matrix which is defined as any square matrix having different diagonal elements and the rest of the elements are 0 . this type of matrix is also called principal diagonal matrix and leading diagonal matrix also.
• scalar matrix:- any square matrix having the same diagonal elements and the rest elements are 0 except 1. 
• identify matrix:- any square matrix having all the elements 1 at its diagonals and rest of the elements are 0 except 1 and represented by me.... when by I2, I3.
• triangular matrix:- this type of matrix is divided into two different matrixes:- 1. upper triangular matrix             2. lower triangular matrix upper triangular matrix:- any square matrix having lower elements are 0 and its diagonal matrix makes a triangle with upper triangle elements.
• lower triangular matrix:- any square matrix having upper elements are 0 and its diagonal matrix with lower elements.
• comparable matrix:- any two matrixes of the same order are known as this matrix, can have different elements.
• equal matrix:- any two matrices are of the same order and have the same elements embedded in them.
• negative matrix:- any matrix having the same and opposite elements of given elements are called a negative matrix.
• let A be a matrix of the order 3*4. if R denote the first row of a and C2 is the second column. then determine the order of R1 and C2.
• addition and subtracting of two matrixes:- addition is like and it occurs only when the square matrix and same order and column also.  just we need to plus the element of one + element of the second matrix. same applied with subtraction of matrixes scalar multiplication: it is defined as the multiplication of a certain number whether it's anything into a matrix. this multiplication is done by multiplying any element given to all the elements of that matrix to get a new multiplied matrix.
• matrix multiplication is the multiplication defined as the multiplication of each element of the first matrix to each element of the other but each element of the first matrix in order wise to the other elements of the second matrix in column-wise.
• this multiplication can be defined as:-
• cube root of unity: 1. only 1           2. w(this is called omega )     3. w2 (this omega square )
• properties of the cube root of unity:- @ the sum of the cube root of unity are 0. 1+w+w2= 0.
• @ these are some results that can be used in questions easily:- 1+w = -w2
•                                                                                                      w2 + w = -1 and 1+w2 = -w 
• these are interrelated to each other.
• @ product of cube root of unity is always  1 . only 
• w = complex root of unity 
• in this matrixes, the principle of mathematical induction is used:- as 
• 1+2+3+4..................+n = n(n+1)/2. this can be understood by only ques that are given search.
• transpose of the matrix:- the transpose of the matrix is defined as the interchanging of rows and columns as the rows are changed and written as columns and columns are changed and written as rows.
• properties of transpose :- # (A+B)t = At +Bt , where t = transpose . #same for A-B # same for multiplication # (At)t =A.
• symmetric matrix:- this type of matrix is defined as the matrix which is transposed and remains the same matrix itself.
• we can say that Aij = A Ji that means rows and columns are the same.
• skew-symmetric matrix:- in which - or negative sign is included in it. as At = -A , Aij = - Aji .
• cofactor of the matrix:- these are to be done in form of steps as step1. CROSS MULTIPLY step2. TRANSPOSE IT .step 3. CHANGE THE SIGNS EXCEPT FOR THE FIRST AND THIRD ELEMENTS OF COLUMN 1 AND THIRD COLUMN.
• adjoint of the matrix is defined as the set of cofactors. the formula of this type is adjA= cofactor of A/determinant of A.
• singular matrix:- this matrix has its determinant as 0 .then we can say that our matrix is singular.
• properties of inverse :- (AB) -1 = B -1 A-1 where -1 is the inverse .
• adj A = A/ IAI,   where inverse can be found as adj A / IAI.
• now elementary row transformations:- this refers to as transformation of row with the help of the formula A=AI.
• elementary column transformation:- this is defined as the transformation of the column with the help of the formula A= IA.
• please make sure that the A in row transformation comes left side, whereas, in column transformation, it comes to the left side.

ok, so guys this is to clear u that some of the topics can only be understood by doing questions or explaining them. I'll be dropping questions and a set of examples to the telegram channel as well as the blog so that u can access it easily.

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