576 CHAPTER 9 Mathematical Systems 2. Identity: Let a b c d e f ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ represent any 2 3 × matrix. Next, consider the 2 3 × matrix 0 0 0 0 0 0 . ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Note that a b c d e f a b c d e f 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ and a b c d e f a b c d e f 0 0 0 0 0 0 + = . ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Therefore, 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ is the identity matrix under matrix addition, called the additive identity matrix. 3. Inverse Elements: Again, let a b c d e f ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ represent any 2 3 × matrix. Next, consider the matrix formed by taking the opposite of each element within this matrix, a b c d e f . − − − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Note that a b c d e f a b c d e f 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + − − − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ and a b c d e f a b c d e f = 0 0 0 0 0 0 . − − − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Therefore, every 2 3 × matrix has an additive inverse. 4. Associative Property: It is given that the associative property holds. One example that illustrates the associative property follows. 1 3 1 2 5 0 0 4 3 7 2 6 2 5 0 3 2 8 1 3 1 2 5 0 0 4 3 7 2 6 2 5 0 3 2 8 1 1 2 5 3 6 2 5 0 3 2 8 1 3 1 2 5 0 2 1 3 4 0 14 3 4 2 2 5 14 3 4 2 2 5 14 ? ? − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 5. Commutative Property: Let a b c d e f ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ and g h i j k l ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ represent any two 2 3 × matrices. Note that a b c d e f g h i j k l a g b h c i d j e k f l ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + + + + + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ and g h i j k l a b c d e f g a h b i c j d k e l f ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + + + + + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Learning Catalytics Keyword: Angel-SOM-9.4 (See Preface for additional details.)
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