If , then value of A–1 is equal to

Engineering

Mathematics

Algebra of Matrices

Inverse of a Matrix

Question

If $A = [\begin{array}{ccc} 3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1 \end{array}]$ , then value of A^–1 is equal to

A²

A³

A⁴

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

$A^{- 1} = \frac{adj (A)}{| A |}; | A | = 3 (1) + 3 (2) + 4 (- 2) = 3 + 6 - 8 = 1$

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

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

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

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

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

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