If and An = 0 then the minimum value of n is

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Question

If $A = [\begin{array}{cc} ab & b^{2} \\ - a^{2} & - ab \end{array}]$ and Aⁿ = 0 then the minimum value of n is

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$A = [\begin{array}{cc} ab & b^{2} \\ - a^{2} & - ab \end{array}]$

A^{2} = [\begin{array}{cc} ab & b^{2} \\ - a^{2} & - ab \end{array}] [\begin{array}{cc} ab & b^{2} \\ - a^{z} & - ab \end{array}] = [\begin{array}{cc} a^{2} b^{2} - a^{2} b^{2} & {ab}^{3} - {ab}^{3} \\ - a^{3} b + a^{3} b & - a^{2} b^{2} + a^{2} b^{2} \end{array}]

= [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]

A² = 0

∴ n = 2.