Multiply the Matrices [[5^n,2^n],[25^n,22^n]][[1,-1],[-2,1]]

Problem

[[5,2],[2⋅5,2⋅2]]*[[1,−1],[−2,1]]

Solution

1. Identify the rule for matrix multiplication, where the element in row i and column j is the dot product of the ith row of the first matrix and the jth column 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.

5*(1)+2*(−2)=5−2⋅2

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.

5*(−1)+2*(1)=−5+2

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.

(2⋅5)*(1)+(2⋅2)*(−2)=2⋅5−4⋅2

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.

(2⋅5)*(−1)+(2⋅2)*(1)=−2⋅5+2⋅2

6. Simplify the expressions using exponent rules where 2⋅2=2(n+1) and 4⋅2=2(n+2)

5−2(n+1)

−5+2

2⋅5−2(n+2)

−2⋅5+2(n+1)

Final Answer

[[5,2],[2⋅5,2⋅2]]*[[1,−1],[−2,1]]=[[5−2(n+1),2−5],[2⋅5−2(n+2),2(n+1)−2⋅5]]

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