If S=0 1 11 0 11 1,A=12b+cc−ab−ac−bc+aa−bb−ca−ca+b, then SAS−1=

S=011101110⇒det⁡S=0(0−1)−1(0−1)+1(1−0) =0+1+1 =2The cofactors of elements of A are A11=+(0−1)=−1,A12=−(0−1)=1,A13=+(1−0)=1A21=−1(0−1)=1,A22=+(0−1)=−1,A23=−(0−1)=1A31=+(1−0)=1,A32=−(0−1)=1,A33=+(0−1)=−1Adj⁡A=A11A21A31A12A22A32A13A23A33=−1111−1111−1∴A−1=AdjAdet⁡A=12−1111−1111−1SAS−1=01110111012b+cc−ab−ac−bc+aa−bb−ca−ca+b12−1111−1111−1=140+c−b+b−c0+c+a+a−c0+a−b+a+bb+c+0+b−ac−a+0+a−cb−a+0+a+bb+c+c−b+0c−a+c+a+0b−a+a−b+0−1111−1111−1=1402a2a2b02b2c2c0−1111−1111−1=140+2a+2a0−2a+2a0+2a−2a−2b+0+2b2b+0+2b2b+0−2b−2c+2c+02c−2c+02c+2c+0=144a0004b0004c=a 0 00 b 00 0 c

If S=0 1 11 0 11 1,A=12b+cc−ab−ac−bc+aa−bb−ca−ca+b, then SAS−1=

S=011101110⇒det⁡S=0(0−1)−1(0−1)+1(1−0) =0+1+1 =2The cofactors of elements of A are A11=+(0−1)=−1,A12=−(0−1)=1,A13=+(1−0)=1A21=−1(0−1)=1,A22=+(0−1)=−1,A23=−(0−1)=1A31=+(1−0)=1,A32=−(0−1)=1,A33=+(0−1)=−1Adj⁡A=A11A21A31A12A22A32A13A23A33=−1111−1111−1∴A−1=AdjAdet⁡A=12−1111−1111−1SAS−1=01110111012b+cc−ab−ac−bc+aa−bb−ca−ca+b12−1111−1111−1=140+c−b+b−c0+c+a+a−c0+a−b+a+bb+c+0+b−ac−a+0+a−cb−a+0+a+bb+c+c−b+0c−a+c+a+0b−a+a−b+0−1111−1111−1=1402a2a2b02b2c2c0−1111−1111−1=140+2a+2a0−2a+2a0+2a−2a−2b+0+2b2b+0+2b2b+0−2b−2c+2c+02c−2c+02c+2c+0=144a0004b0004c=a 0 00 b 00 0 c

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$If S = [\begin{array}{lll} 0 1 1 \\ 1 0 1 \\ 1 1 \end{array}], A = \frac{1}{2} [\begin{array}{ccc} b + c & c - a & b - a \\ c - b & c + a & a - b \\ b - c & a - c & a + b \end{array}], then S A S^{- 1} =$

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$[\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}]$

$\frac{1}{2} [\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}]$

$2 [\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}]$

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Detailed Solution

$\begin{array}{l} S = [\begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}] \\ \Rightarrow \det S = 0 (0 - 1) - 1 (0 - 1) + 1 (1 - 0) \\ = 0 + 1 + 1 \\ = 2 \end{array}$

The cofactors of elements of A are

$\begin{array}{l} A_{11} = + (0 - 1) = - 1, A_{12} = - (0 - 1) = 1, A_{13} = + (1 - 0) = 1 \\ A_{21} = - 1 (0 - 1) = 1, A_{22} = + (0 - 1) = - 1, A_{23} = - (0 - 1) = 1 \\ A_{31} = + (1 - 0) = 1, A_{32} = - (0 - 1) = 1, A_{33} = + (0 - 1) = - 1 \\ Adj A = [\begin{array}{ccc} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{array}] = [\begin{array}{ccc} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{array}] ∴ A^{- 1} = \frac{A d j A}{\det A} = \frac{1}{2} [\begin{array}{ccc} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{array}] \end{array}$

$\begin{array}{l} S A S^{- 1} = [\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}] \frac{1}{2} [\begin{array}{ccc} b + c & c - a & b - a \\ c - b & c + a & a - b \\ b - c & a - c & a + b \end{array}] \frac{1}{2} [\begin{array}{ccc} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{array}] \\ = \frac{1}{4} [\begin{array}{ccc} 0 + c - b + b - c & 0 + c + a + a - c & 0 + a - b + a + b \\ b + c + 0 + b - a & c - a + 0 + a - c & b - a + 0 + a + b \\ b + c + c - b + 0 & c - a + c + a + 0 & b - a + a - b + 0 \end{array}] [\begin{array}{ccc} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{array}] \\ = \frac{1}{4} [\begin{array}{ccc} 0 & 2 a & 2 a \\ 2 b & 0 & 2 b \\ 2 c & 2 c & 0 \end{array}] [\begin{array}{ccc} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{array}] \\ = \frac{1}{4} [\begin{array}{ccc} 0 + 2 a + 2 a & 0 - 2 a + 2 a & 0 + 2 a - 2 a \\ - 2 b + 0 + 2 b & 2 b + 0 + 2 b & 2 b + 0 - 2 b \\ - 2 c + 2 c + 0 & 2 c - 2 c + 0 & 2 c + 2 c + 0 \end{array}] \\ = \frac{1}{4} [\begin{array}{ccc} 4 a & 0 & 0 \\ 0 & 4 b & 0 \\ 0 & 0 & 4 c \end{array}] \end{array}$