Multiply the Matrices

Problem

[[a,b,c],[b,c,a],[c,a,b]]*[[−a,c,b],[−b,a,c],[−c,b,a]]

Solution

1. Identify the dimensions of the matrices. Both are 3×3 matrices, so their product will also be a 3×3 matrix.

2. Calculate the element in the first row, first column by taking the dot product of the first row of the first matrix and the first column of the second matrix.

a*(−a)+b*(−b)+c*(−c)=−a2−b2−c2

3. Calculate the element in the first row, second column.

a(c)+b(a)+c(b)=a*c+a*b+b*c

4. Calculate the element in the first row, third column.

a(b)+b(c)+c(a)=a*b+b*c+a*c

5. Calculate the element in the second row, first column.

b*(−a)+c*(−b)+a*(−c)=−a*b−b*c−a*c

6. Calculate the element in the second row, second column.

b(c)+c(a)+a(b)=b*c+a*c+a*b

7. Calculate the element in the second row, third column.

b(b)+c(c)+a(a)=b2+c2+a2

8. Calculate the element in the third row, first column.

c*(−a)+a*(−b)+b*(−c)=−a*c−a*b−b*c

9. Calculate the element in the third row, second column.

c(c)+a(a)+b(b)=c2+a2+b2

10. Calculate the element in the third row, third column.

c(b)+a(c)+b(a)=b*c+a*c+a*b

11. Assemble the resulting matrix using the calculated values.

Final Answer

[[a,b,c],[b,c,a],[c,a,b]]*[[−a,c,b],[−b,a,c],[−c,b,a]]=[[−(a2+b2+c2),a*b+b*c+a*c,a*b+b*c+a*c],[−(a*b+b*c+a*c),a*b+b*c+a*c,a2+b2+c2],[−(a*b+b*c+a*c),a2+b2+c2,a*b+b*c+a*c]]

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