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 September 27th, 2015, 09:23 PM #1 Member   Joined: Jan 2012 Posts: 51 Thanks: 1 Question about invertible matrices Need to show that if A is not invertible then exist B (nxn) such that AB=0 but B \= 0. Thought about writing like A=E1...Ek*R where Ei are elementary matrices and R(different from I) the row reduced echelon matrice but doesnt seem.to help, any idea?? Last edited by bonildo; September 27th, 2015 at 09:37 PM. September 29th, 2015, 10:10 AM #2 Member   Joined: Jan 2012 Posts: 51 Thanks: 1 anyone ?? September 30th, 2015, 04:37 PM #3 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Consider a matrix $A_{n\times n}$ which is not invertible. Then there exist $v=\left(a_1,\ldots,a_n\right)\in\mathbb R^n$ such that $v\neq 0$ and $Av=0$. Consider the matrix $B_{n\times n}=\left[\begin{array}{cccc}a_1 & a_1 & \ldots & a_1 \\a_2 & a_2 & \ldots & a_2\\\vdots & \vdots & \ldots & \vdots \\a_n & a_n & \ldots & a_n \end{array}\right]$. Since $v\neq 0$ we have that $B\neq 0$. We also have that $Be_i=v$ where $e_i=\left(0,0,\ldots,\underbrace{1}_{i^{th}}, \ldots,0\right)$ for $1\leq i\leq n$.\\ Consider a generic vector $w=\displaystyle\sum_{i=1}^n\alpha_ie_i$ for some $\alpha_i\in\mathbb R$. We have that $Bw=B\left(\displaystyle\sum_{i=1}^{n}\alpha_i e_i\right)=\displaystyle\sum_{i=1}^{n}\alpha_i Be_i=\displaystyle\sum_{i=1}^{n}\alpha_i v=\left(\displaystyle\sum_{i=1}^{n}\alpha_i\right) v$.\\ Hence $ABw=A\left(\left(\displaystyle\sum_{i=1}^n\alpha_ i\right)v\right)=\left(\displaystyle\sum_{i=1}^{n} \alpha_i\right)Av=0$ which give us that $AB=0$ for every vector in $\mathbb R^n$. Tags invertible, matrices, question Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Colonel Mustard Applied Math 3 May 8th, 2013 01:14 PM Jascha Linear Algebra 3 February 21st, 2012 03:00 PM Riazy Linear Algebra 1 January 3rd, 2012 07:56 AM Steel Linear Algebra 2 December 4th, 2010 03:20 PM TsAmE Linear Algebra 12 November 2nd, 2010 09:52 AM

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