Find the inverse of by column transformation.

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Find the inverse of $A = [\begin{array}{ccc} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}]$ by column transformation.

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Take A = Al

[\begin{array}{ccc} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}] = A [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

c₂ → c₂ – 3c₁ and c₃ → c₃ – 3c₁

[\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}] = A [\begin{array}{cc} 1 & - 3 & - 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

c₁ → c₁ – c₂

[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}] = A [\begin{array}{ccc} 4 & - 3 & - 3 \\ - 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

c₁ → c₁ – c₃

[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] = A [\begin{array}{ccc} 7 & - 3 & - 3 \\ - 1 & 0 & 0 \\ - 1 & 0 & 1 \end{array}]

∴ A^{- 1} = [\begin{array}{ccc} 7 & - 3 & - 3 \\ - 1 & 0 & 0 \\ - 1 & 0 & 1 \end{array}]

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