Multiply the Matrices [[-1,3^(n+1)],[1,3^n]][[1,3],[-1,-1]]

Problem

[[−1,3(n+1)],[1,3]]*[[1,3],[−1,−1]]

Solution

1. Identify the matrix multiplication rule where the element in row i and column j of the resulting matrix is the dot product of row i of the first matrix and column j of the second matrix.

2. Calculate the first row, first column entry by multiplying the first row of the left matrix by the first column of the right matrix.

(−1)*(1)+(3(n+1))*(−1)=−1−3(n+1)

3. Calculate the first row, second column entry by multiplying the first row of the left matrix by the second column of the right matrix.

(−1)*(3)+(3(n+1))*(−1)=−3−3(n+1)

4. Calculate the second row, first column entry by multiplying the second row of the left matrix by the first column of the right matrix.

(1)*(1)+(3)*(−1)=1−3

5. Calculate the second row, second column entry by multiplying the second row of the left matrix by the second column of the right matrix.

(1)*(3)+(3)*(−1)=3−3

6. Combine these entries into the resulting 2×2 matrix.

Final Answer

[[−1,3(n+1)],[1,3]]*[[1,3],[−1,−1]]=[[−1−3(n+1),−3−3(n+1)],[1−3,3−3]]

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