Properties of Matrix Operations

Matrix Addition

The fundamental properties of real number addition also apply to matrices.

Let \( A \), \( B \), and \( C \) be \( m \times n \) matrices:

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

Matrix Multiplication

Unlike addition, not all multiplication properties of real numbers apply to matrices.

• Matrix multiplication is not commutative: Even if both \( AB \) and \( BA \) are defined, they may not be equal.
• A matrix may not have a multiplicative inverse, even if it is square.

However, some properties do generalize. Let \( A \), \( B \), and \( C \) be matrices such that the operations are defined:

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: \( I_m A = A \), \( A I_n = A \)

Scalar Multiplication

Let \( r \), \( s \) be real numbers, and \( A \), \( B \) be matrices:

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 defined)

Transpose of a Matrix

Let \( r \) be real, and \( A \), \( B \) be matrices:

1. \( (A^T)^T = A \)
2. \( (A + B)^T = A^T + B^T \)
3. \( (AB)^T = B^T A^T \)
4. \( (rA)^T = rA^T \)

Determinants

1. \( \det(A) = \det(A^T) \)
2. If any row/column is multiplied by \( k \), then \( \det(\Delta') = k \cdot \det(\Delta) \)
3. If a row or column is all zeros, then \( \det = 0 \)
4. If matrix is upper/lower triangular, \( \det = \) product of diagonal elements