➤ Addition and
Subtraction of matrices
A + B = B + A
(A + B) + C= A +
(B + C)
k (A + B)=kA + kB
➤ Multiplication of
Matrices
AB BA
(AB)C =A(BC)
Distributive law
A (B + C) = AB +
AC
(A + B) C =AC+BC
Multiplicative identity for a square Matrix A;
AI = IA
= A
➤ Properties of
Transpose of Matrix
(Aᵀ)ᵀ =
A
(k A)ᵀ = k Aᵀ
(A+B)ᵀ = Aᵀ + Bᵀ
(A B)ᵀ = Bᵀ Aᵀ
➤ Symmetric and Skew
Symmetric Matrices
Symmetric Matrix
–if Aᵀ = A
Skew Symmetric
Matrix– If Aᵀ = - A
Note: In Skew matrix diagonal element are always 0
For any square
matrix A,
(A+Aᵀ)
is a symmetric matrix
(A–Aᵀ)
is a skew symmetric matrix
➤ Properties of
inverse
1. For matrix A, A⁻¹ is unique, i.e. there is only one inverse of a matrix
2. (A⁻¹)⁻¹ = A
3. (kA) = 1/k A⁻¹
4.(A⁻¹)ᵀ =(Aᵀ)⁻¹
5. (A + B)⁻¹ =A⁻¹ + B⁻¹
6. (AB)⁻¹ = B⁻¹ A⁻¹
➤ Properties
of Determinants
1.
Determinant of any identity matrix is 1
or I AI = 1
2.
I ATI =I A I
3.
IABI =IAI IBI
4.
IA⁻¹I =1/ IAI
5.
IkAI = kⁿ IAI where n is
order of the matrix
6.
Similarly I –A I = I –1× A I
=(–1)ⁿ × IAI
7.
(adj A) A =A (adj A) = IAI I
8. Determinant of adj A;
Iadj AI = IAI ; where n is order of the matrix
