If then find (AT)

Engineering

Mathematics

Introduction to Matrix

Inverse of a Matrix

Transpose and Adjoint of a Matrix

Question

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

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

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

${(A^{⊤})}^{- 1} = \frac{1}{|A^{⊤}|} adj [A^{⊤}]$

|A^T| = 1(– 1 – 8) – 0 – 2(– 8 + 3)

– 9 + 10 ....(1)

Cofactor $adj A = [\begin{array}{ccc} - 9 & + 8 & - 5 \\ - 8 & 7 & - 4 \\ - 2 & + 2 & - 1 \end{array}]$

So, ${(A^{⊤})}^{- 1} = \frac{1}{1} [\begin{array}{lll} - 9 8 - 5 \\ - 8 7 - 4 \\ - 2 2 - 1 \end{array}]$