If , find (A')

Engineering

Mathematics

Introduction to Matrix

Transpose and Adjoint of a Matrix

Inverse of a Matrix

Question

If $A = [\begin{array}{ccc} 1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1 \end{array}]$ , find (A')^–1.

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$A = [\begin{array}{ccc} 1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1 \end{array}] \to [\begin{array}{lll} 1 0 0 \\ 0 1 0 \\ 0 1 0 \end{array}]$

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

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

$[\begin{array}{ccc} - 1 & - 2 & 3 \\ 0 & 1 & 1 \\ 1 & 0 & 4 \end{array}] = [\begin{array}{lll} 1 0 0 \\ 1 1 0 \\ 0 2 1 \end{array}] \Rightarrow [\begin{array}{ccc} - 1 & 0 & 5 \\ 0 & 1 & 1 \\ - 1 & 0 & 4 \end{array}] = [\begin{array}{lll} 3 2 0 \\ 1 1 0 \\ 0 2 1 \end{array}]$

$\Rightarrow [\begin{array}{ccc} - 1 & 0 & 5 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}] = [\begin{array}{lll} 3 2 0 \\ 1 1 0 \\ 3 0 - 1 \end{array}] \Rightarrow [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}] = [\begin{array}{lll} - 3 0 1 \\ 1 1 0 \\ 0 2 1 \end{array}]$

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