Properties of Matrix Operations

Matrix Addition

The properties of real number addition extend naturally to matrices. Let A, B, and C be m × n matrices:

1. Commutative Property: A + B = B + A
2. Associative Property: A + (B + C) = (A + B) + C
3. Additive Identity: There exists a unique zero matrix Om × n such that A + Om × n = A
4. Additive Inverse: For every matrix A, there exists -A such that A + (-A) = Om × n

Matrix Multiplication

Matrix multiplication differs from scalar multiplication: not all properties of real numbers hold. Let A, B, and C be matrices with compatible dimensions for multiplication:

• Non-Commutative: In general, AB ≠ BA, even if both products are defined.
• Existence of Inverse: A square matrix may not have a multiplicative inverse.

Some properties do hold for matrix multiplication:

1. Associative Property: A(BC) = (AB)C
2. Left Distributive Property: A(B + C) = AB + AC
3. Right Distributive Property: (A + B)C = AC + BC
4. Multiplicative Identity: Im A = A, A In = A, where Im and In are identity matrices of appropriate dimensions

An identity matrix In is an n × n square matrix with ones on the diagonal and zeros elsewhere.

Scalar Multiplication

Let r and s be real numbers, and A, B matrices of compatible dimensions:

1. r(sA) = (rs)A
2. (r + s)A = rA + sA
3. r(A + B) = rA + rB
4. A(rB) = r(AB) = (rA)B (if matrix multiplication is defined)

Transpose of a Matrix

The transpose of a matrix A, denoted AT, is formed by swapping its rows and columns. That is, the element in the i-th row and j-th column of A becomes the element in the j-th row and i-th column of AT.

For any matrices A, B and scalar r:

1. (AT)T = A
2. (A + B)T = AT + BT
3. (AB)T = BT AT
4. (rA)T = r AT

Determinants

The determinant is a scalar associated with a square matrix that indicates whether the matrix is invertible and provides information about volume scaling and system behavior. It can be calculated recursively using cofactor expansion (see the Minor, Cofactor, and Adjoint section) or, for small matrices, by simple formulas. For example, for a 2 × 2 matrix:

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

the determinant is calculated as:

\[ \det(A) = ad - bc \]

1. det(AB) = det(A) · det(B)
2. det(A) = det(AT)
3. If a row or column is multiplied by a scalar k, the determinant is scaled: det(Δ') = k · det(Δ)
4. If a row or column consists entirely of zeros, then det(Δ) = 0
5. If Δ is upper or lower triangular, det(Δ) equals the product of its diagonal elements