Simplify the Matrix

Problem

[[−√(,2)/2,−√(,2)/2,(4√(,3))/3],[√(,2)/2,0,(4√(,3))/3],[0,√(,2)/2,(4√(,3))/3]]*[[−1/√(,2),1/√(,2),0],[−1/√(,2),0,1/√(,2)],[1/√(,3),1/√(,3),1/√(,3)]]

Solution

1. Identify the matrices to be multiplied. Let the first matrix be A and the second matrix be B We will calculate the product C=A*B where each element (c_i*j) is the dot product of the ith row of A and the jth column of B

2. Calculate the elements of the first row of the resulting matrix:

(c_11)=(−√(,2)/2)*(−1/√(,2))+(−√(,2)/2)*(−1/√(,2))+((4√(,3))/3)*(1/√(,3))=1/2+1/2+4/3=7/3

(c_12)=(−√(,2)/2)*(1/√(,2))+(−√(,2)/2)*(0)+((4√(,3))/3)*(1/√(,3))=−1/2+0+4/3=5/6

(c_13)=(−√(,2)/2)*(0)+(−√(,2)/2)*(1/√(,2))+((4√(,3))/3)*(1/√(,3))=0−1/2+4/3=5/6

3. Calculate the elements of the second row of the resulting matrix:

(c_21)=(√(,2)/2)*(−1/√(,2))+(0)*(−1/√(,2))+((4√(,3))/3)*(1/√(,3))=−1/2+0+4/3=5/6

(c_22)=(√(,2)/2)*(1/√(,2))+(0)*(0)+((4√(,3))/3)*(1/√(,3))=1/2+0+4/3=11/6

(c_23)=(√(,2)/2)*(0)+(0)*(1/√(,2))+((4√(,3))/3)*(1/√(,3))=0+0+4/3=4/3

4. Calculate the elements of the third row of the resulting matrix:

(c_31)=(0)*(−1/√(,2))+(√(,2)/2)*(−1/√(,2))+((4√(,3))/3)*(1/√(,3))=0−1/2+4/3=5/6

(c_32)=(0)*(1/√(,2))+(√(,2)/2)*(0)+((4√(,3))/3)*(1/√(,3))=0+0+4/3=4/3

(c_33)=(0)*(0)+(√(,2)/2)*(1/√(,2))+((4√(,3))/3)*(1/√(,3))=0+1/2+4/3=11/6

Final Answer

[[−√(,2)/2,−√(,2)/2,(4√(,3))/3],[√(,2)/2,0,(4√(,3))/3],[0,√(,2)/2,(4√(,3))/3]]*[[−1/√(,2),1/√(,2),0],[−1/√(,2),0,1/√(,2)],[1/√(,3),1/√(,3),1/√(,3)]]=[[7/3,5/6,5/6],[5/6,11/6,4/3],[5/6,4/3,11/6]]

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