Let’s look at various properties of Matrices and Determinants
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
 (A ^{ T } ) ^{ T } = A
 (kA) ^{ T } = kA ^{ T }
 (A + B) ^{ T } = A ^{ T } + B ^{ T }
 (AB) ^{ T } = B ^{ T } A ^{ T }
Symmetric and Skew Symmetric matrices
 Symmetric Matrix  If A ^{ T } = A

Skew  symmetric Matrix  If A
^{
T
}
= A
Note: In a skew matrix, diagonal elements are always 0 . 
For any square matrix A,
(A + A ^{ T } ) is a symmetric matrix
(A − A ^{ T } ) is a skewsymmetric matrix
Inverse of a matrix
For a square matrix A, if
AB = BA = I
Then, B is the inverse of A
i.e. B = A
^{
−1
}
We will find inverse of a matrix by
Properties of Inverse

For a matrix A,
A ^{ −1 } is unique, i.e., there is only one inverse of a matrix  (A ^{ −1 } ) ^{ −1 } = A

(𝑘 𝐴)
^{
−1
}
= 1/𝑘 𝐴
^{
−1
}
Note: This is different from
(kA) ^{ T } = k A ^{ T }  (A ^{ 1 } ) ^{ T } = (A ^{ T } ) ^{ 1 }
 (A+B) ^{ 1 } = A ^{ 1 } + B ^{ 1 }
 (𝐴𝐵) ^{ −1 } =𝐵 ^{ −1 } 𝐴 ^{ −1 }
Important things to note in Determinants

Determinant of Identity matrix = 1
det (I) = 1  A ^{ T }  = A
 AB = A B
 A ^{ −1 }  = 1/𝐴
 kA = k ^{ n } A where n is order of matrix

Similarly,
−A = −1 × A
= (−1)n × A  (adj A) A = A (adj) = AI

Deteminant of adj A
"adj A = A  ^{ 𝑛−1 } where n is the order of determinant
Number multiplied to matrix and determinant
Other important points
Also, look at
 Finding Inverse using Elementary Transformation
 Finding Inverse using Adjoint
 Properties of Determinant

Difference between Matrices and Determinants