Simplify the Matrix

Problem

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

Solution

1. Multiply the first matrix by the diagonal matrix. Since the second matrix is diagonal, each column of the first matrix is scaled by the corresponding diagonal entry.

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

2. Multiply the resulting matrix by the third matrix. Calculate the entry at row 1, column 1.

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

3. Calculate the entry at row 1, column 2.

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

4. Calculate the entry at row 1, column 3.

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

5. Calculate the entry at row 2, column 1.

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

6. Calculate the entry at row 2, column 2.

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

7. Calculate the entry at row 2, column 3.

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

8. Calculate the entry at row 3, column 1.

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

9. Calculate the entry at row 3, column 2.

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

10. Calculate the entry at row 3, column 3.

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

Final Answer

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