This document covers matrices, determinants, and their properties. Some key points: - A matrix is a rectangular arrangement of numbers. The order of a m×n matrix is defined by its m rows and n columns. - A square matrix is symmetric if it is equal to its transpose, and skew-symmetric if it is equal to the negative of its transpose. - The determinant of a square matrix provides a number associated with the matrix, denoted |A| or detA. - Properties of determinants include that the value does not change under row/column interchange operations, and that multiplying a row/column by a constant k multiplies the determinant by k.

1. CHAPTERS – 3 & 4
MATRICES AND DETERMINANTS
 A matrix is an rectangular arrangement of numbers.
 If a matrix has m rows and n columns then its order is m×n.
 Two matrices A and B are equal if their order is same and corresponding elements
must havesame values.
 A squarematrix is symmetric if A’= A and a squarematrix is skew symmetric if A’=-A.
 Any squarematrix can be expressed as a sum of symmetric and skew symmetric
matrix.
 If A and B are two squarematrices such that AB= BA= I, then B is called inverse of A
and denoted by A-1
and A is inverseof B.
 Elementary transformations (operations) areas follows;
o Ri↔ Rj or Ci↔Cj.
o Ri→ kRi or Ci → kCj
o Ri→Ri+kRj or Ci →Ci+kCj.
 To every squarematrix A wecan associate a number called determinant of the matrix
A and is denoted by |A| or detA
 Area of triangle with vertices (x1, y1) , (x2,y2) , (x3,y3) is given by ∆=
1
2
|
𝑥1 𝑦1 1
𝑥2 𝑦2 1
𝑥3 𝑦3 1
|
 Minor of an element aij of the determinant of matrix A is the determinant obtained by
deleting its ith row and jth column and denoted by Mij.
 Cofactor of aij is given by Aij =(-1)i+j Mij.
 A squarematrix A is said to be singular if |A|=0 and non singular if |A|≠0
 The adjoint of a squarematrix A is defined as the transposeof the matrix of cofactors.
Itis denoted by adj A
 A squarematrix A is invertible only if matrix is non singular and A-1
=
1
|𝐴|
adj A

2.  If A is a squarematrix of order n , then |kA|=kn
|A|
 If A is a squarematrix of order n , then |adjA|= |A|n-1
 If A is a squarematrix of order n , then |A.adjA|= |A|n
 A(adjA) =(adjA).A =|A|I
 For a systemof linear equations in AX=B if
o If |A|=0, then systemis consistent and has unique solution X=A-1B.
o If |A|≠0 and adjA. B ≠0, then systemis inconsistentand has no solution.
o If |A|≠0 and adjA. B =0, then systemis consistentand has infinitely many
solution.
 Properties of determinants:
 The value of determinant remains unchanged if we interchange its rows and columns.
 If any two rows or columns of a determinants are interchange , then value of
determinant changes by minus sign.
 If wemultiply each element of a row or column of a determinant by constant k, then
value of determinant gets multiplied by k.
 If elements of row or column in a determinant are expressed as a sumof two or more
terms, then the given determinants can be expressedas a sumof two or more
determinants.
o The value of determinant remains unchanged if we apply the operations
o Ri→Ri+kRj or Ci →Ci+kCj.