1.3 Properties of Matrix operations

# 1.3 Properties of Matrix operations - Properties of Matrix...

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Properties of Matrix operations. Theorem 1. Properties of Matrix Addition For m × n matrices A , B , and C , the following properties hold. Closure property: A + B is again an m × n matrix. Associative property: ( A + B ) + C = A + ( B + C ) . Commutative property: A + B = B + A . Additive identity: The m × n matrix 0 consisting of all zeros has the property that A + 0 = A . Additive inverse: The m × n matrix ( A ) has the property that A + ( A ) = 0 . Theorem 2. Distributive and Associative Laws For conformable matrices each of the following is true. A ( B + C ) = AB + AC (left-hand distributive law). ( D + E ) F = DF + EF (right-hand distributive law). A ( BC ) = ( AB ) C (associative law) Example 1. Let A = [ 2 2 3 3 1 2 ] ,B = [ 1 0 2 2 3 1 ] ,C = [ 1 2 1 0 2 2 ] . Then A ( B + C ) = [ 2 2 3 3 1 2 ] [ 0 2 3 2 5 3 ] = [ 21 1 7 2 ] and AB + AC = [ 2 2 3 3 1 2 ] [ 1 0 2 2 3 1 ] + [ 2 2 3 3 1 2 ] [ 1 2 1 0 2 2 ] = ¿

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¿ [ 15 1 7 4 ] + [ 6 2 0 2 ] = [ 21 1 7 2 ] Definition. The n×n scalar matrix I n = [ 10 .0 01 .0 ………… 00 .1 ] is called the identity matrix of order n.
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