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Evaluate $\det A$ for an $n\times n$ matrix $A=[a_{ij}]$ when

(1) $a_{ij}= 1$ if $i\ne j$, $a_{ij}=0$ if $i=j$

(2) $a_{ij} = i+2j$

I study linear algebra alone. I don't have a man asking math questions around me :( I'm a little hazy on this problem. First I use cofactor expansion. However I am very confusing about $n \times n$ matrix. Mealwhile I think use Math induction. Can you help me??

Minus One-Twelfth
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Ben
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  • For the first one, see https://math.stackexchange.com/questions/84206/how-to-calculate-the-following-determinants-all-ones-minus-i. – Minus One-Twelfth Mar 29 '19 at 08:15
  • You can only use induction if you already have the expression for the determinant, to prove that it holds in general. You can't use induction to find the expression itself. If you are able to derive the expression, induction won't be necessary anymore. – Shubham Johri Mar 29 '19 at 08:22
  • Oh, I can't find out a while ago. I'm not used to this website. Thank you so much :) – Ben Mar 29 '19 at 08:27
  • hmm, so i can't use induction.. thank you for your help – Ben Mar 29 '19 at 08:29
  • The second part is easy if you notice that the rows and columns of the matrix for $n>2$ are arithmetic progressions. For $n>2$, perform the operations $R_n\to R_n-R_{n-1}$ and $R_{n-1}\to R_{n-1}-R_{n-2}$ in that order, getting two identical rows. For $n=1,2$, the determinant is non-zero and can be calculated separately. – Shubham Johri Mar 29 '19 at 08:33
  • thanks so much :) I will try it! – Ben Mar 29 '19 at 10:24

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