Matrices

Matrices
A Matrix is an array of real numbers.

Equality of Matrices
If two matrices are equal then their corresponding elements are equal.

For two matrices to b equal their order has to be exactly same.

Algebra of Matrices
When Matrices are added or subtracted, their corresponding elements are added or subtracted.

Transpose of a Matrix
Transpose of a matrix is a matrix of inverted rows and columns.

Determinant of a Matrix
Determinant of Matrix gives the magnitude of the matrix.

Inverse of a Matrix
Inverse of an identity matrix is an identity matrix again.

Properties of Matrices

 * 1) AA-1 = I
 * 2) (A-1)-1 = A
 * 3) (AT)T = A
 * 4) (AB)-1 = B-1A-1
 * 5) (AB)T =  BTAT
 * 6) (A + B)T = AT + BT (If addition subtraction possible)
 * |A2| = |A|2
 * 1) (Ak)-1 = (A-1)k
 * 2) (AT)-1 = (A-1)T
 * 3) Do Adjoint first or any other operation first, the answer is same.
 * 4) (Adj A )m= (Adj Am)
 * 5) (Adj A)T = (Adj AT)
 * 6) (Adj A)-1 = (Adj A-1)
 * 1) (Adj A)T = (Adj AT)
 * 2) (Adj A)-1 = (Adj A-1)

Useful Formulae

 * 1) (kA)-1 = 1/k A-1
 * 2) A.(Adj A) = (Adj A)A = |A| = In
 * |Adj A| = |A|n-1
 * 1) adj(adj A) = |A|n-2. A
 * 2) adj (kA) = kn-1 (Adj A)

Other Types of Matrices

 * 1) Orthogonal Matrix : If ATA = AAT = I  ; i.e. Transpose is the inverse of the matrix . If A is an orthogonal Matrix, the A-1 is also an orthogonal matrix.

Tips and Tricks
1) Prefer Adjoint Method over row transformations.