Simplify the Matrix

Problem

[[−9,8],[1,1]]*[[−34,0],[0,34]]*[[−1,8],[1,9]]

Solution

1. Multiply the first two matrices by scaling the columns of the first matrix by the diagonal entries of the second.

[[−9,8],[1,1]]*[[−34,0],[0,34]]=[[(−9)*(−34),(8)*(34)],[(1)*(−34),(1)*(34)]]

2. Simplify the entries of the resulting matrix using the property (−9)*(−34)=9⋅34 and (1)*(−34)=−34

[[9⋅34,8⋅34],[−34,34]]

3. Multiply this result by the third matrix using the standard row-by-column rule.

[[9⋅34,8⋅34],[−34,34]]*[[−1,8],[1,9]]=[[(9⋅34)*(−1)+(8⋅34)*(1),(9⋅34)*(8)+(8⋅34)*(9)],[(−34)*(−1)+(34)*(1),(−34)*(8)+(34)*(9)]]

4. Combine like terms for each entry of the matrix.

[[−9⋅34+8⋅34,72⋅34+72⋅34],[34+34,−8⋅34+9⋅34]]

5. Simplify the arithmetic coefficients to reach the final form.

[[−34,144⋅34],[2⋅34,34]]

Final Answer

[[−9,8],[1,1]]*[[−34,0],[0,34]]*[[−1,8],[1,9]]=[[−34,144⋅34],[2⋅34,34]]

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