Chapter 3: Matrices (Final Corrected)

Operations, Properties & Logic Rules

1. Properties of Transpose ($$A’$$)

Double Transpose $$(A’)’ = A$$
Scalar Multiple $$(kA)’ = kA’$$
Sum/Difference Rule $$(A \pm B)’ = A’ \pm B’$$
Reversal Law $$(AB)’ = B’A’$$
Power Transpose $$(A^n)’ = (A’)^n$$

2. Symmetric and Skew-Symmetric Matrices

Symmetric Definition $$A’ = A$$
Skew-Symmetric Definition $$A’ = -A$$ (Diagonal is zero)
Symmetric Part $$\frac{1}{2}(A + A’)$$ is always symmetric
Skew-Symmetric Part $$\frac{1}{2}(A – A’)$$ is always skew-symmetric
Decomposition Theorem $$A = \frac{1}{2}(A + A’) + \frac{1}{2}(A – A’)$$
Both S & SS Property If $$A’=A$$and$$A’=-A$$$$\Rightarrow$$$$A = O$$ (Null Matrix)

3. Properties of Matrix Addition

Commutative Law $$A + B = B + A$$
Associative Law $$(A + B) + C = A + (B + C)$$
Additive Identity $$A + O = A = O + A$$
Additive Inverse $$A + (-A) = O$$
Cancellation Law $$A + B = A + C \Rightarrow B = C$$

4. Properties of Matrix Multiplication

Non-Commutative $$AB \neq BA$$ (In general)
Associative Law $$(AB)C = A(BC)$$
Distributive Law $$A(B + C) = AB + AC$$
Multiplicative Identity $$AI = IA = A$$
Multiplicative Inverse $$A \cdot A^{-1} = I$$(if$$|A| \neq 0$$)
Cancellation Law (Caution) $$AB = AC \Rightarrow B = C$$(Only if$$|A| \neq 0$$)
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