How to Start Linear Algebra

• Square matrix:- If the number of rows of a matrix is equal to the number of columns of a matrix , then is called a square matrix.
• i.e If order of Matrix is m x n then m=n for square matrix.
• Upper triangular matrix: – A matrix for which is called an upper triangular matrix. That is, all the elements below the diagonal entries are zero.
• Lower triangular matrix: – A matrix for which is called a lower triangular matrix. That is, all the elements above the diagonal entries are zero.
• Diagonal matrix: – A square matrix with all non-diagonal elements equal to zero is called a diagonal matrix, that is, only the diagonal entries of the square matrix can be non-zero,
• Identity matrix: A diagonal matrix with all diagonal elements equal to one is called an identity matrix.
• Zero matrix:- A matrix whose all entries are zero is called a zero matrix
• Transpose of a matrix: Interchanging of Rows and Columns of matrix is known as transpose of Matrix. The transpose of A is denoted by
• Symmetric matrix:-
• skew-symmetric matrix:- all the diagonal elements of a skew-symmetric matrix have to be zero.
• Trace of a matrix:- Sum of diagonal elements of matrix is known as Trace of matrix.

Determinant of any square matrix:-
Determinant of any Matrix can be obtained by addition of product of elements with their respective Co-fators of any Row(Column).

• Rank of a matrix :- The rank of a matrix is defined as the order of the largest square submatrix whose determinant is not zero.
• Consistent and inconsistent system of equations :- A system of equations is consistent if there is a solution, and it is inconsistent if there is no solution. However, a consistent system of equations does not mean a unique solution, that is, a consistent system of equations may have a unique solution or infinite solutions. [A][X]=[B] is a system of Linear Equation.
• [A] is square matrix, [ A:B] is a augmented Matrix and [B] is a column Matrix.

• Consistency check for Homogenous equation:- Equations which are of the [A][X]=[0] are known as homogenous equation.
• Homogenous eqautions are always consistent, so only two conditions occurs:-
• When |A| = 0 then Infinte many solutions or Non-Trival solution
• When |A| not equals to Zero then unique solution or trival solution.
• Inverse of a matrix:- If [A] is any square matrix of order (nxn) then, will be the Inverse of [A] .
• Such that
• and A-1= Adj [A] Where Adj.[A] = Adjoint of Matrix [A] & |A| = Determinant of Matrix [A].
• Adjoint of Matix [A] is obtained by :-
• First replace all elements by their respective Co-Factors
• Then take transpose of the matrix, the resultant will be Adjoint of [A]