Simplify the Matrix

Problem

(([3,1/2,5],[1,−1,3])+([2,−5/2,0],[0,2,−3]))*([1,0,−1],[1,2,0],[0,3,0])

Solution

1. Add the matrices inside the parentheses by summing corresponding elements.

([3+2,1/2−5/2,5+0],[1+0,−1+2,3−3])=([5,−2,5],[1,1,0])

2. Set up the multiplication of the resulting 2×3 matrix and the 3×3 matrix.

([5,−2,5],[1,1,0])*([1,0,−1],[1,2,0],[0,3,0])

3. Calculate the first row of the product by taking the dot product of the first row of the left matrix with each column of the right matrix.

(R_1)*(C_1)=(5)*(1)+(−2)*(1)+(5)*(0)=3

(R_1)*(C_2)=(5)*(0)+(−2)*(2)+(5)*(3)=11

(R_1)*(C_3)=(5)*(−1)+(−2)*(0)+(5)*(0)=−5

4. Calculate the second row of the product by taking the dot product of the second row of the left matrix with each column of the right matrix.

(R_2)*(C_1)=(1)*(1)+(1)*(1)+(0)*(0)=2

(R_2)*(C_2)=(1)*(0)+(1)*(2)+(0)*(3)=2

(R_2)*(C_3)=(1)*(−1)+(1)*(0)+(0)*(0)=−1

Final Answer

(([3,1/2,5],[1,−1,3])+([2,−5/2,0],[0,2,−3]))*([1,0,−1],[1,2,0],[0,3,0])=([3,11,−5],[2,2,−1])

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