Find the Determinant of the Resulting Matrix [[1,2,3],[x,y,z]][[1,x],[2,y],[3,z]]

Problem

det(([1,2,3],[x,y,z])*([1,x],[2,y],[3,z]))

Solution

1. Multiply the two matrices to find the resulting 2×2 matrix.

A=([1,2,3],[x,y,z])

B=([1,x],[2,y],[3,z])

C=A*B=([1*(1)+2*(2)+3*(3),1*(x)+2*(y)+3*(z)],[x(1)+y(2)+z(3),x(x)+y(y)+z(z)])

2. Simplify the entries of the resulting matrix C

C=([1+4+9,x+2*y+3*z],[x+2*y+3*z,x^2+y^2+z^2])

C=([14,x+2*y+3*z],[x+2*y+3*z,x^2+y^2+z^2])

3. Apply the determinant formula for a 2×2 matrix, which is det(C)=a*d−b*c

det(C)=(14)*(x^2+y^2+z^2)−(x+2*y+3*z)*(x+2*y+3*z)

4. Expand the squared trinomial term.

(x+2*y+3*z)^2=x^2+4*y^2+9*z^2+4*x*y+6*x*z+12*y*z

5. Subtract the expanded term from the first term and combine like terms.

det(C)=14*x^2+14*y^2+14*z^2−(x^2+4*y^2+9*z^2+4*x*y+6*x*z+12*y*z)

det(C)=13*x^2+10*y^2+5*z^2−4*x*y−6*x*z−12*y*z

Final Answer

det(([1,2,3],[x,y,z])*([1,x],[2,y],[3,z]))=13*x^2+10*y^2+5*z^2−4*x*y−6*x*z−12*y*z

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