# Exercise 3 .pdf - Math 3A c 2022 March Boedihardjo...

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Math 3AExerciseMarch Boedihardjo c 2022Important note:The questions below vary a lot in difficulty. For the difficult questions, be patientwhen thinking about them. You will learn much more if you spend hours and hours thinking andget nothing before looking at the solution than if you look at the solution directly without thinking.Notation: IfAis a square matrix thenAkis the productA . . . A|{z}k. For example,A2=AA.Iis the identity matrix.1. Write down a 2×2 matrixA(with real entries) such thatA10=IandAk6=Ifor all 1k9.2. Find matricesCandDsuch thatC1325=2-312and1325D=2-312.3. Suppose thatA-1=1231andB-1=2113. Find a vectorvsuch thatABv=10.4. Suppose thatA=abc3is not invertible andA11=12. Finda, b, c.5. Suppose thatAis a 2×2 invertible matrix such thatA45=11andA13=10.FindA-113.6. Suppose thatAis a 2×2 rotation matrix such thatA2a=13wherea >0. Finda.7.(i) Show that every 2×2 rotation matrix is invertible.(ii) Show that ifAis a 2×2 rotation matrix thenA+12Iis invertible.8. Which of the following 2×2 matrices are rotation matrices?(i)1001(ii)-100-1(iii)0-110(iv)3-443(v)153-443(vi)1534
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