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Consider the Matrix M: M= a b       c d  First, find the determinant. To do this, we use the formula:  det(M) = ad - bc Then plug the information into the formula used to find the inverse: 1/det(m) x   c ...

Proof by induction

Base Case when n=1 

LHS = 1^3=1 RHS= 1/4(1^2)(1+1)^2=1/4(1)(2^2)=1/4(4)

Assume true for n=k 

so  ∑r^3= 1/4k^2(k+1)^2

For n=k+1 

∑r^3 = ∑k terms...

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This qustion can be solved easily using the gradient formular, m = ∆y/∆x, and some simple algrebra.

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We can expand following any row or column we want. A wise choice would be to use a row/column in which we have one or more zero entries to reduce the calculations. From this, we pick the first entry on th...