If [tex]A=\left[\begin{array}{cc}3 & 5 \\ 1 & -9\end{array}\right][/tex] and [tex]B=\left[\begin{array}{ll}0 & 4 \\ 6 & 3\end{array}\right][/tex] show that:
1) [tex]A+B=B+A[/tex]
ii) [tex]2 A-2 B=2(A-B)[/tex].
Answer (2)
We have demonstrated that matrix addition is commutative by showing A + B = B + A and also verified that the distributive property of scalar multiplication over matrix subtraction holds, establishing 2 A − 2 B = 2 ( A − B ) . Both properties illustrate fundamental aspects of matrix operations. Therefore, the calculations confirm the desired results of the operations on matrices A and B.
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Verify that matrix addition is commutative by showing that A + B = B + A .
Calculate A + B = [ 3 7 9 − 6 ] .
Calculate B + A = [ 3 7 9 − 6 ] .
Verify that scalar multiplication distributes over matrix subtraction by showing that 2 A − 2 B = 2 ( A − B ) , which is [ 6 − 10 2 − 24 ] .
Explanation
Problem Analysis We are given two matrices, A and B , and we need to show two properties: 1) that matrix addition is commutative, meaning A + B = B + A , and 2) that scalar multiplication distributes over matrix subtraction, meaning 2 A − 2 B = 2 ( A − B ) .
Part 1: Proving A + B = B + A First, let's prove that A + B = B + A . We will compute A + B and B + A separately and show that they are equal.
Calculating A + B Compute A + B :
A + B = [ 3 1 5 − 9 ] + [ 0 6 4 3 ] = [ 3 + 0 1 + 6 5 + 4 − 9 + 3 ] = [ 3 7 9 − 6 ]
Calculating B + A Compute B + A :
B + A = [ 0 6 4 3 ] + [ 3 1 5 − 9 ] = [ 0 + 3 6 + 1 4 + 5 3 + ( − 9 ) ] = [ 3 7 9 − 6 ]
Verifying A + B = B + A Since A + B = [ 3 7 9 − 6 ] and B + A = [ 3 7 9 − 6 ] , we have shown that A + B = B + A .
Part 2: Proving 2A - 2B = 2(A - B) Now, let's prove that 2 A − 2 B = 2 ( A − B ) . We will compute 2 A − 2 B and 2 ( A − B ) separately and show that they are equal.
Calculating 2A Compute 2 A :
2 A = 2 [ 3 1 5 − 9 ] = [ 2 ( 3 ) 2 ( 1 ) 2 ( 5 ) 2 ( − 9 ) ] = [ 6 2 10 − 18 ]
Calculating 2B Compute 2 B :
2 B = 2 [ 0 6 4 3 ] = [ 2 ( 0 ) 2 ( 6 ) 2 ( 4 ) 2 ( 3 ) ] = [ 0 12 8 6 ]
Calculating 2A - 2B Compute 2 A − 2 B :
2 A − 2 B = [ 6 2 10 − 18 ] − [ 0 12 8 6 ] = [ 6 − 0 2 − 12 10 − 8 − 18 − 6 ] = [ 6 − 10 2 − 24 ]
Calculating A - B Compute A − B :
A − B = [ 3 1 5 − 9 ] − [ 0 6 4 3 ] = [ 3 − 0 1 − 6 5 − 4 − 9 − 3 ] = [ 3 − 5 1 − 12 ]
Calculating 2(A - B) Compute 2 ( A − B ) :
2 ( A − B ) = 2 [ 3 − 5 1 − 12 ] = [ 2 ( 3 ) 2 ( − 5 ) 2 ( 1 ) 2 ( − 12 ) ] = [ 6 − 10 2 − 24 ]
Verifying 2A - 2B = 2(A - B) Since 2 A − 2 B = [ 6 − 10 2 − 24 ] and 2 ( A − B ) = [ 6 − 10 2 − 24 ] , we have shown that 2 A − 2 B = 2 ( A − B ) .
Final Answer In conclusion, we have shown that for the given matrices A and B , A + B = B + A and 2 A − 2 B = 2 ( A − B ) .
Examples
Matrix operations are fundamental in computer graphics for transformations like scaling, rotation, and translation of objects. For instance, when designing a video game, matrices are used to manipulate the position and orientation of characters and objects in 3D space. The commutative property of matrix addition ensures that combining multiple translations in different orders yields the same final position. Similarly, scalar multiplication is used to uniformly scale objects, and the distributive property ensures that scaling and then translating is equivalent to translating and then scaling.
