If A -1-2-221-22-21- show that adj A-3 AT

A= 1 2 221 22 21Now,C11=1 2 21= 3, #160;C12= 2 221= 6 #160;and #160;C13=212 2= 6C21=–2 2 21=6, #160;C22= 1 221=3 #160;and #160;C23=–1 22 2= 6C31= 2 ...

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Standard XII

Chemistry

Chemical Properties of Amines

If A =-1-2-22...

Question

If $A = [\begin{matrix} - 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1 \end{matrix}]$ , show that adj A = 3A^T.

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Solution

$A = [\begin{matrix} - 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1 \end{matrix}] Now, C_{11} = |\begin{matrix} 1 & - 2 \\ - 2 & 1 \end{matrix}| = - 3, C_{12} = - |\begin{matrix} 2 & - 2 \\ 2 & 1 \end{matrix}| = - 6 and C_{13} = |\begin{matrix} 2 & 1 \\ 2 & - 2 \end{matrix}| = - 6 C_{21} = - |\begin{matrix} - 2 & - 2 \\ - 2 & 1 \end{matrix}| = 6, C_{22} = |\begin{matrix} - 1 & - 2 \\ 2 & 1 \end{matrix}| = 3 and C_{23} = - |\begin{matrix} - 1 & - 2 \\ 2 & - 2 \end{matrix}| = - 6 C_{31} = |\begin{matrix} - 2 & - 2 \\ 1 & - 2 \end{matrix}| = 6, C_{32} = - |\begin{matrix} - 1 & - 2 \\ 2 & - 2 \end{matrix}| = - 6 and C_{33} = |\begin{matrix} - 1 & - 2 \\ 2 & 1 \end{matrix}| = 3 adj A = {[\begin{matrix} - 3 & - 6 & - 6 \\ 6 & 3 & - 6 \\ 6 & - 6 & 3 \end{matrix}]}^{T} = [\begin{matrix} - 3 & 6 & 6 \\ - 6 & 3 & - 6 \\ - 6 & - 6 & 3 \end{matrix}] A^{T} = [\begin{matrix} - 1 & 2 & 2 \\ - 2 & 1 & - 2 \\ - 2 & - 2 & 1 \end{matrix}] \Rightarrow 3 A^{T} = [\begin{matrix} - 3 & 6 & 6 \\ - 6 & 3 & - 6 \\ - 6 & - 6 & 3 \end{matrix}] \Rightarrow 3 A^{T} = adj A$

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If A=-1-2-221-22-21, show that adj A = 3AT.

A=-1-2-221-22-21Now,C11=1-2-21=-3, C12=-2-221=-6 and C13=212-2=-6C21=--2-2-21=6, C22=-1-221=3 and C23=--1-22-2=-6C31=-2-21-2=6, C32=--1-22-2=-6 and C33=-1-221=3adj A=-3-6-663-66-63T=-366-63-6-6-63AT=-122-21-2-2-21⇒3AT=-366-63-6-6-63⇒3AT=adj A

If $A = [\begin{matrix} - 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1 \end{matrix}]$ , show that adj A = 3A^T.