SOME TYPES OF MATRICES--DEFINITION AND PROPERTIES

Symmetric Matrix
A square matrix A= [aij] is symmetric if aij = aji - for every i and j
A= A^T
Skew Symmetric Matrix
A square matrix A=[aij ] is skew symmetric if aij = -aji . for every i and j
A= -A^T
Orthogonal Matrix
A square matrix A is orthogonal if AA^T = A^TA = I

Important Theorems And Properties

1. Every sq. matrix can be expressed as the sum of a symmetric and skew symmetric matrix
2. Inverse of a non-singular symmetric matrix A is symmetric
3. If A is a symmetric matrix then adj(A) is also symmetric
4. If A,B are orthogonal matrices then AB and BA are orthogonal
5. Inverse of orthogonal matrix is orthogonal and its transpose is also orthogonal
6. A,B are invertible matrices then (AB)^-1 = (B)^-1(A)^-1

IMPORTANT QUESTIONS:

Q1.show that 1/2 mat (A) is orthogonal where mat(A) is
-1 1 1 1
1-1 1 1
1 1-1 1
1 1 1-1
soln : Find A^T
By Definition AA^T = I
Solve for LHS (the ans. is a 4x4 unit matrix)
Hence LHS = RHS


Q2. show that A³ = A^-1 where A is
3 -3 4
2 -3 4
0 -1 1

soln : solve A² and assume as B
then solve AB
to solve A^-1
A^-1 = adj (A) / A
to solve adj A
the cofactor for every aij is to be taken
cofactor, is (-1)i+j Mij where Mij is minor of every aij