Simplify the Matrix

Problem

[[1,4],[−2,3],[1,−2]]*[[3,6],[6,0]]*[[3,0,0],[0,6,−3]]

Solution

1. Multiply the first two matrices. Let A be the 3×2 matrix and B be the 2×2 matrix.

A×B=[[(1)*(3)+(4)*(6),(1)*(6)+(4)*(0)],[(−2)*(3)+(3)*(6),(−2)*(6)+(3)*(0)],[(1)*(3)+(−2)*(6),(1)*(6)+(−2)*(0)]]

2. Simplify the resulting 3×2 matrix.

A×B=[[3+24,6+0],[−6+18,−12+0],[3−12,6+0]]

A×B=[[27,6],[12,−12],[−9,6]]

3. Multiply the result by the third 2×3 matrix. Let C be the third matrix.

(A×B)×C=[[27,6],[12,−12],[−9,6]]*[[3,0,0],[0,6,−3]]

4. Calculate the final matrix elements.

(A×B)×C=[[(27)*(3)+(6)*(0),(27)*(0)+(6)*(6),(27)*(0)+(6)*(−3)],[(12)*(3)+(−12)*(0),(12)*(0)+(−12)*(6),(12)*(0)+(−12)*(−3)],[(−9)*(3)+(6)*(0),(−9)*(0)+(6)*(6),(−9)*(0)+(6)*(−3)]]

5. Simplify the final values.

(A×B)×C=[[81+0,0+36,0−18],[36+0,0−72,0+36],[−27+0,0+36,0−18]]

Final Answer

[[1,4],[−2,3],[1,−2]]*[[3,6],[6,0]]*[[3,0,0],[0,6,−3]]=[[81,36,−18],[36,−72,36],[−27,36,−18]]

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