Multiply the Matrices [[-2/3],[2/3],[1/3]][[-2/3,2/3,1/3]]

Problem

[[−2/3],[2/3],[1/3]]*[[−2/3,2/3,1/3]]

Solution

1. Identify the dimensions of the matrices. The first matrix is a 3×1 column vector and the second is a 1×3 row vector. The resulting product will be a 3×3 matrix.

2. Calculate the first row of the product matrix by multiplying the first element of the column vector, −2/3 by each element of the row vector.

(−2/3)*(−2/3)=4/9

(−2/3)*(2/3)=−4/9

(−2/3)*(1/3)=−2/9

3. Calculate the second row of the product matrix by multiplying the second element of the column vector, 2/3 by each element of the row vector.

(2/3)*(−2/3)=−4/9

(2/3)*(2/3)=4/9

(2/3)*(1/3)=2/9

4. Calculate the third row of the product matrix by multiplying the third element of the column vector, 1/3 by each element of the row vector.

(1/3)*(−2/3)=−2/9

(1/3)*(2/3)=2/9

(1/3)*(1/3)=1/9

5. Assemble the calculated values into the final 3×3 matrix.

Final Answer

[[−2/3],[2/3],[1/3]]*[[−2/3,2/3,1/3]]=[[4/9,−4/9,−2/9],[−4/9,4/9,2/9],[−2/9,2/9,1/9]]

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