A matrix is a rectangular arrange of numbers arranged in rows and columns. Many operations may be performed on matrices.

NOTATION OF A MATRIX:

Addition of matrices:

3 by 3 matrix

2 by 2 matrix

Subtraction of matrices

2 by 2 matrix

3  by 3 matrix

Scalar multiplication:

 for all

Properties of scalar multiplication:

• K(A + B) = KA + KB
• (K + l)A = KA + lA
• (Kl)A = K(lA) = l(KA)
• (-K)A = -(KA) = K(-A)
• 1·A = A
• (-1)A = -A

Multiplication of matrices:

2 by 2 matrix

 3 by 3 matrix

Properties of multiplication if matrices:

• AB ≠ BA
• A(BC) = (AB)C
• A(B + C) = AB + AC
• (A + B)C = AC + BC

Transpose of a matrix

noted as A^T

Properties:

• AT)T = A
• (A + B)T = AT + BT, A and B being of the same order.
• (KA)T= KAT, K is any scalar 
• (AB)T= BTAT, A and B being conformable for the product AB. (This is also called reversal law.)

Trace of a matrix:

its the sum of its diagonal properties:

RULES OF TRACES:

• tr(AB) = tr(BA)
• tr(A) = tr(AT)
• tr(cA) = c tr(A), for a scalar 'c'
• tr(A + B) = tr(A) + tr(B)

Determinant of a matrix:

|A|=det(A)

 3 by 3 matrix

Det(A)

2 by 2 matrix

Minor of a matrix:

 using this, you can find the matrix of M

Cofactor of a matrix:

see for yourself...

Adjoint of a matrix:

adj(A) = C^T

or the transpose of the cofactor of a matrix

Inverse of a matrix: