Chapter 3 – Matrices

Order of a Matrix

If a matrix has m rows and n columns, its order is m × n.

Matrix Addition and Subtraction

A + B = [a_ij + b_ij]
A - B = [a_ij - b_ij]

Scalar Multiplication

kA = [ka_ij]

Matrix Multiplication

C = A · B, where c_ij = Σ(a_ik · b_kj)
condition: Number of columns in A = Number of rows in B

Transpose of a Matrix

A^T_ij = [a_ji]

Properties of Transpose

(A^T)^T = A
(A + B)^T = A^T + B^T
(kA)^T = kA^T
(AB)^T = B^TA^T

Symmetric and Skew-Symmetric Matrices

A matrix A is symmetric if A = A^T
A matrix A is skew-symmetric if A^T = -A

Identity Matrix

A matrix I is an identity matrix if IA = AI = A

Zero Matrix Properties

A matrix O is a zero matrix if all its elements are zero.
kO = O for any scalar k.

Matrices