x-y+z=4 x+y+z=2 2x+y-3z=0

{tex}A = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&1 \\ 2&1&{ - 3} \\ 1&1&1 \end{array}} \right]{/tex}

{tex}\left| A \right| = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&1 \\ 2&1&{ - 3} \\ 1&1&1 \end{array}} \right]{/tex}

{tex} = 10 \ne 0{/tex}

Here,
A₁₁ = 4, A₁₂ = -5, A₁₃ = 1
A₂₁ = 2, A₂₂ = 0, A₂₃ = -2
A₃₁ = 2, A₃₂ = 5, A₃₃ = 3

{tex}adjA = \left[ {\begin{array}{*{20}{c}} 4&2&2 \\ { - 5}&0&5 \\ 1&{ - 2}&3 \end{array}} \right]{/tex}

{tex}{A^{ - 1}} = \frac{1}{{\left| A \right|}}\left( {adjA} \right){/tex}

{tex}= \frac{1}{{10}}\left[ {\begin{array}{*{20}{c}} 4&2&2 \\ { - 5}&0&5 \\ 1&{ - 2}&3 \end{array}} \right]{/tex}

System of equation can be written is

X = A^-1B

{tex} = \frac{1}{{10}}\left[ {\begin{array}{*{20}{c}} 4&2&2 \\ { - 5}&0&5 \\ 1&{ - 2}&3 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 4 \\ 0 \\ 2 \end{array}} \right]{/tex}

{tex}\left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right]{/tex}

x = 2, y = -1, z = 1