If possible, using elementary row transformations, find the inverse of the following matrix.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

If possible, using elementary row transformations, find the inverse of the following matrix.
$[\begin{array}{ccc} 2 & 3 & - 3 \\ - 1 & - 2 & 2 \\ 1 & 1 & - 1 \end{array}]$

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As, A = IA,

∴ [\begin{array}{ccc} 2 & 3 & - 3 \\ - 1 & - 2 & 2 \\ 1 & 1 & - 1 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] A

And we know that I = A^–1A.

So, by row transformation we will try to convert LHS into I

[\begin{array}{ccc} ∵ R_{2} \to & R_{2} + & R_{3} \\ and R_{1} \to & R_{1} - & 2 R_{3} \end{array}]

\Rightarrow [\begin{array}{ccc} 0 & 1 & - 1 \\ 0 & - 1 & 1 \\ 1 & 1 & - 1 \end{array}] = [\begin{array}{ccc} 1 & 0 & - 2 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}] A

[∵ R₂ → R₂ + R₁]

\Rightarrow [\begin{array}{ccc} 0 & 1 & - 1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{array}] = [\begin{array}{ccc} 1 & 0 & - 2 \\ 2 & 1 & - 2 \\ 0 & 0 & 1 \end{array}] A

Since, second row of the matrix A on LHS is containing all zeroes, so we can say that every inverse of matrix A does not exist.