Matrices and matrix arithmetic Let M2(R) represent the set of all 2 x 2 matrices with entries from the real numbers, let the operation + represent matrix addition, and let the operation * represent matrix multiplication. (Matrix addition and matrix multiplication: if ?= [? ?? ?] and ?= [? ?? ℎ], then ?+?= [?+ ? ?+ ??+ ? ?+ ℎ]; and ?∗?= [??+ ?? ??+ ?ℎ??+ ?? ??+ ?ℎ]) a. For the system (M2(R), +), demonstrate or explain why: i. (M2(R), +) is closed ii. (M2(R), +) is associative (work out the algebra to prove this) iii. (M2(R), +) has an identity element, i.e., an additive identity iv. Each element in M2(R) has an additive inverse v. (M2(R), +) is commutative
Matrices and matrix arithmetic
Let M2(R) represent the set of all 2 x 2 matrices with entries from the real
numbers, let the operation + represent matrix addition, and let the operation
* represent matrix multiplication.
(Matrix addition and matrix multiplication:
if ?= [? ?? ?] and ?= [? ?? ℎ], then ?+?= [?+ ? ?+ ??+ ? ?+ ℎ];
and ?∗?= [??+ ?? ??+ ?ℎ??+ ?? ??+ ?ℎ])
a. For the system (M2(R), +), demonstrate or explain why:
i. (M2(R), +) is closed
ii. (M2(R), +) is associative (work out the algebra to prove this)
iii. (M2(R), +) has an identity element, i.e., an additive identity
iv. Each element in M2(R) has an additive inverse
v. (M2(R), +) is commutative
Step by step
Solved in 4 steps with 4 images
For the system (M2(R), *), demonstrate or explain why:
i. (M2(R), *) is closed
ii. (M2(R), *) is associative (work out the algebra to prove this)
c. For the system (M2(R), *), demonstrate or explain why the operation *
distributes over the operation +, i.e. work out the algebra to prove that
for any matrices A, B and C from M2(R):
i. A*(B + C) = A*B + A*C, and
ii. (B + C)*A = B*A + C*A
by Bartleby Expertd. Demonstrate or explain why the system (M2(R), *) has an identity
element.
e. Demonstrate that each element in the system (M2(R), *) does not
necessarily have a (multiplicative) inverse.
f. Demonstrate why the system (M2(R), *) is not commutative.
by Bartleby Expert
