Suppose A is a matrix such that $A^{2} = A$ and $(I + A)^{10} = I + k A$ , find $k$ .

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Solution

Given $A^{2} = A$
$(I + A)^{10} = I + k A$
$^{10} C_{0} I +^{10} C_{1} A +^{10} C_{2} A^{2} +^{10} C_{3} A^{3} + . . . .^{10} C_{10} A^{10} = I + k A$
$\Rightarrow I + (10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1) A = I + k A$
$\Rightarrow k = 1023$

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