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# Matrix_3 - Matrix Properties Objectives At the end of this...

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Matrix Properties © Arizona State University, Department of Mathematics and Statistics 1 of 4 Subtraction is always anti-commutative even in the real numbers. Objectives: At the end of this lesson, you should be able to: 1. List the matrix properties. 2. Know when they cannot apply! 3. Understand the matrix identity. 4. Be aware of matrix weirdness. Background You already seen examples of differences in the properties of matrix multiplication from the real numbers. We have proof that when a product is even defined sometimes . Imagine the AB BA pain if you had to learn the times table in two different ways in the third grade! The Properties However, many properties do carry through. Lets review them. The matrices are A, B, and C. When the matrix sizes are compatible as shown in the last lesson, we have these properties: Commutativity of Addition: A B B A + = + Anti-Commutativity of Subtraction: ( ) A B B A - = - - Associativity of Addition: . ( ) ( ) A B C A B C A B C + + = + + = + + Associativity of Multiplication: ( ) ( ) A BC AB C ABC = = Scalar Multiplication: For k any real number, for , 11 1 1 n m mn a a A a a = vertellipsis downslopeellipsis vertellipsis midhorizellipsis 11 1 1 n m mn ka ka kA ka a = vertellipsis downslopeellipsis vertellipsis midhorizellipsis Scalar Distribution: For m and n any real numbers: and ( ) mA nA m n A + = + . ( ) m A B mA mB + = + Matrix Distributions: Note the plural. When we can do the indicated multiplications and additions, specifically in the order they are shown, we have both a left and right distribution property: ( ) ( ) A B C AB AC and B C A BA CA + = + + = + Properties of Zero: For the appropriately-sized zero matrix , and . 0 0

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