Simplify the Matrix

Problem

[[2,0],[7,−1]]*[[6,−1,3],[5,0,−2]]+[[7,0],[−1,3]]*[[6,−1,3],[5,0,−2]]

Solution

1. Identify the common matrix factor on the right side of both terms in the expression.

2. Apply the distributive property of matrix multiplication over addition, which states A*C+B*C=(A+B)*C

3. Add the two matrices inside the parentheses.

[[2,0],[7,−1]]+[[7,0],[−1,3]]=[[2+7,0+0],[7−1,−1+3]]

[[2+7,0+0],[7−1,−1+3]]=[[9,0],[6,2]]

4. Multiply the resulting sum by the common matrix.

[[9,0],[6,2]]*[[6,−1,3],[5,0,−2]]

5. Calculate the entries of the product matrix using row-by-column multiplication.

Row 1: *[9*(6)+0*(5),9*(−1)+0*(0),9*(3)+0*(−2)]=[54,−9,27]

Row 2: *[6*(6)+2*(5),6*(−1)+2*(0),6*(3)+2*(−2)]=[46,−6,14]

Final Answer

[[2,0],[7,−1]]*[[6,−1,3],[5,0,−2]]+[[7,0],[−1,3]]*[[6,−1,3],[5,0,−2]]=[[54,−9,27],[46,−6,14]]

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