Without expanding- prove thatabcxyzpqr - xyzpqrabc - ybqxapzcr

Xyzpqrabc #160;R2 #8596;R3= xyzabcpqr #160;R1 #8596;R2=abcxyzpqrybqxapzcr #160;=yxzbacqpr #160;C1 #8596;C2= xyzabcpqr #160;R1 #8596;R2=abcxyzp ...

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Change of Variables

Without expan...

Question

Without expanding, prove that

$|\begin{matrix} a & b & c \\ x & y & z \\ p & q & r \end{matrix}| = |\begin{matrix} x & y & z \\ p & q & r \\ a & b & c \end{matrix}| = |\begin{matrix} y & b & q \\ x & a & p \\ z & c & r \end{matrix}|$

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∣ ∣ ∣ ∣ \begin{matrix} a & b & c x & y & z p & q & r \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} y & b & q x & a & p z & c & r \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} x & y & z p & q & r a & b & c \end{matrix} ∣ ∣ ∣ ∣

Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c x & y & z p & q & r \end{matrix} ∣ ∣ ∣ ∣

Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} y & b & q x & a & p z & c & r \end{matrix} ∣ ∣ ∣ ∣

Δ_{1}

Without expanding the determinant, prove that

∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & b c b & b^{2} & c a c & c^{2} & a b \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a^{2} & a^{3} 1 & b^{2} & b^{3} 1 & c^{2} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & b c b & b^{2} & c a c & c^{2} & a b \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a^{2} & a^{3} 1 & b^{2} & b^{3} 1 & c^{2} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣

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