Matrix Operations

Matrix Definitions

• A matrix of dimension (m × 1) is called a row vector.
• A matrix of dimension (1 × n) is called a column vector.

Matrix Operations

• Matrices A and B can be added or subtracted if both have the same dimension (m × n).
• The sum C = A + B is defined as cij = aij + bij.
• The multiplication of a matrix A by a scalar c is defined as cA = [caij].
• The product of an (m × n) matrix A and an (n × p) matrix B results in an (m × p) matrix C.
• Matrix multiplication is not commutative, i.e., AB ≠ BA.

Transpose of a Matrix

• The transpose of a matrix A = [aij] is denoted as AT = [aji].
• If A is of dimension (m × n), then AT is of dimension (n × m).

Diagonal Matrix

• A diagonal matrix is a square matrix with nonzero elements only along the principal diagonal.

Identity Matrix

• The identity (or unit) matrix is a diagonal matrix with 1's along the principal diagonal.

Symmetric Matrix

• A symmetric matrix is a square matrix whose elements satisfy aij = aji or A = AT.

Determinant of a Matrix

• The determinant of a square matrix A is denoted as det A.
• For a 3 × 3 matrix, the determinant is calculated as: det A = a11 (a22 a33 − a23 a32) − a12 (a21 a33 − a23 a31) + a13 (a21 a32 − a22 a31)

Matrix Inversion

• If det A ≠ 0, then A has an inverse, denoted by A^(-1).
• The inverse satisfies the relations A^(-1) A = A A^(-1) = I.
• If det A = 0, then A is singular, and the inverse is not defined.

Gaussian Elimination

• Gaussian elimination is a method of solving simultaneous equations by successively eliminating unknowns.
• It can be used to solve systems of linear equations, such as: 4x1 + 2x2 - 2x3 - 8x4 = 4 x1 + 2x2 + x3 = 2 0.5x1 - x2 + 4x3 + 4x4 = 10 -4x1 - 2x2 - x4 = 0
• The solution can be obtained using Gaussian elimination or Gauss program.