Math 2250-4 Wed Feb 63.4 Matrix algebra\302\267 Our first exam is next Thursday Feb. 14. You have a homework assignment covering 3.5-3.6 which is due Wednesday Feb. 13, and which will be posted on our homework page by later today. (The 3.1-3.4 homework is due this Friday.) The exam will cover through 3.6.\302\267 Today we first discuss the general conclusions about how the shape of the reduced row echelon form of a matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5influences the possible solution sets to linear systems of equations for which it is the coefficient matrix. This is Exercise 6 on Tuesday's notes, which we were in the middle of at the end of class yesterday..............................................................................................\302\267 Then we will discuss vector and matrix algebra, section 3.4. Matrix vector algebra that we've already touched on, but that we want to record carefully:Vector addition and scalar multiplication: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Vector dot product , which yields a scalar (i.e. number) output (regardless of whether vectors are column vectors or row vectors):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY4LUkobWZlbmNlZEdGJDYqLUYjNigtSSdtdGFibGVHRiQ2OS1JJG10ckdGJDYmLUkkbXRkR0YkNigtSSVtc3ViR0YkNiUtSSNtaUdGJDYnUSJ4RicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2KC1JI21uR0YkNiZRIjFGJ0ZARkYvRkpRJ25vcm1hbEYnRkAvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGRS9GR1EqMkR+T3V0cHV0RidGUi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zqbi8lK2dyb3VwYWxpZ25HRmpuLyUocm93c3BhbkdGUS8lK2NvbHVtbnNwYW5HRlFGaG5GW29GXW8tRjQ2Ji1GNzYoLUY6NiVGPC1GIzYoLUZPNiZRIjJGJ0ZARkZGUkZARlRGV0ZZRlJGZW5GaG5GW29GXW9GX29GYW9GaG5GW29GXW8tRjQ2Ji1GNzYoLUY6NiVGPC1GIzYoLUZPNiZRIjNGJ0ZARkZGUkZARlRGV0ZZRlJGZW5GaG5GW29GXW9GX29GYW9GaG5GW29GXW8tRjQ2Ji1GNzYoLUYjNictSSNtb0dGJDYvUSI6RidGQEZGRlIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRmVxLyUpc3RyZXRjaHlHRmVxLyUqc3ltbWV0cmljR0ZlcS8lKGxhcmdlb3BHRmVxLyUubW92YWJsZWxpbWl0c0dGZXEvJSdhY2NlbnRHRmVxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZHJGQEZDRkZGSUZobkZbb0Zdb0Zfb0Zhb0ZobkZbb0Zdby1GNDYmLUY3NigtRjo2JUY8LUYjNigtRj02J1EibkYnRkBGQ0ZGRklGQEZURldGWUZSRmVuRmhuRltvRl1vRl9vRmFvRmhuRltvRl1vLyUmYWxpZ25HUSVheGlzRicvRmluUSliYXNlbGluZUYnL0Zcb1EnY2VudGVyRicvRl5vUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGRS8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRl90LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmp0LyUmZnJhbWVHRmp0LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGZXEvJS1lcXVhbGNvbHVtbnNHRmVxLyUtZGlzcGxheXN0eWxlR0ZlcS8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZndEZARlRGV0ZZRlJGQEZGRlIvSSttc2VtYW50aWNzR0YkUSpDb2xWZWN0b3JGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zddi1GYHE2LlEpJm1pZGRvdDtGJ0ZARlJGY3FGZnFGaHFGanFGXHJGXnJGYHIvRmNyUSwwLjE2NjY2NjdlbUYnL0ZmckZqdi1GLDYqLUYjNigtRjE2OS1GNDYmLUY3NigtRiM2JS1GOjYlLUY9NidRInlGJ0ZARkNGRkZJRkxGZW5GQEZSRmhuRltvRl1vRl9vRmFvRmhuRltvRl1vLUY0NiYtRjc2KC1GIzYlLUY6NiVGandGaW9GZW5GQEZSRmhuRltvRl1vRl9vRmFvRmhuRltvRl1vLUY0NiYtRjc2KC1GIzYlLUY6NiVGandGZHBGZW5GQEZSRmhuRltvRl1vRl9vRmFvRmhuRltvRl1vRmlwLUY0NiYtRjc2KC1GIzYlLUY6NiVGandGXXNGZW5GQEZSRmhuRltvRl1vRl9vRmFvRmhuRltvRl1vRmJzRmVzRmdzRmlzRlt0Rl10RmB0RmJ0RmV0Rmh0Rlt1Rl11Rl91RmJ1RmR1RmZ1Rmh1Rlt2RkBGVEZXRllGUkZARkZGUkZddkZgdkZjdkZddi1GYHE2LlEqJmNvbG9uZXE7RidGRkZSRmNxRmZxRmhxRmpxRlxyRl5yRmByRmJyRmVyLUY6NiUtRj02JkY/RkNGRkZJLUYjNiYtRk82JUZRRkZGUkZDRkZGSUZlbi1GYHE2LlEifkYnRkZGUkZjcUZmcUZocUZqcUZcckZeckZgci9GY3JRJjAuMGVtRicvRmZyRmR6LUY6NiUtRj02JkZceEZDRkZGSUZcekZlbi1GYHE2LlEiK0YnRkZGUkZjcUZmcUZocUZqcUZcckZeckZgci9GY3JRLDAuMjIyMjIyMmVtRicvRmZyRl5bbC1GOjYlRmp5LUYjNiYtRk82JUZdcEZGRlJGQ0ZGRklGZW5GYHotRjo2JUZoekZiW2xGZW5GanotRmBxNi5RIy4uRidGRkZSRmNxRmZxRmhxRmpxRlxyRl5yRmByRl1bbEZlei1GYHE2LlEiLkYnRkZGUkZjcUZmcUZocUZqcUZcckZeckZgckZjekZlekZgekZqekZgei1GOjYlRmp5LUYjNiYtRj02JkZhc0ZDRkZGSUZDRkZGSUZlbkZgei1GOjYlRmh6RmBcbEZlbkZbXGxGUg==Matrix times vector: If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEoJnRpbWVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0Yv matrix and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= column vector, 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 way to write our usual linear system: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 1a): Check that the vector dot product distributes over vector addition and scalar multiplication, i.e.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 .Since the dot product is commutative or by checking directly we also 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 .1b) Use your work from a to show that matrix multiplication distributes over vector addition and scalar multiplication, i.e.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 2 (relates to hw and an example from yesterday): Consider the matrix 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 .Assume the system is consistent, so that there is at least one (particular) solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1GLzYlUSJQRidGNS9GOlEnaXRhbGljRidGNUZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjpRJ25vcm1hbEYn, i.e. a vector LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNiUtRiw2JVEiUEYnRjIvRjdRJ2l0YWxpY0YnLyUwZm9udF9zdHlsZV9uYW1lR1EnTm9ybWFsRicvRjdRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= so 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.Use the algebra properties of Exercise 1b to show that2a) If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1GLzYlUSJIRidGNS9GOlEnaXRhbGljRidGNUZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjpRJ25vcm1hbEYn solves the homogeneous system 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 , then 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 solves the original 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 .2b) Show that every solution to 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 is of the 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJQRidGNS9GOlEnaXRhbGljRidGNS8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRkQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZGL0Y6USdub3JtYWxGJw== is the same single particular solution, and LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNiQtRiw2JVEiSEYnRjIvRjdRJ2l0YWxpY0YnL0Y3USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn is some solution to the homogeneous equation. Remark: The upshot of problem 2 is that if you know all solutions to the homogeneous problem, and you are given a single solution to the non-homogeneous problem, then you actually know all solutions to the non-homogeneous problem.Matrix algebra: \302\267 addition and scalar multiplication: Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm1GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRi82JVEibkYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm1GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRi82JVEibkYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB be two matrices of the same dimensions (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= rows and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= columns). Let 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, 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 . (In this case we write 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 .) Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be a scalar. 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 .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 .In other words, addition and scalar multiplication are defined analogously as for vectors. In fact, for these two operations you can just think of matrices as vectors written in a rectangular rather than row or column format.Exercise 3) Let 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 and 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 . Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiNEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYzNi1RKCZtaW51cztGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGVy1GSzYlUSJCRidGTkZRRi8= .\302\267 matrix multiplication: Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm1GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRi82JVEibkYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm5GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRi82JVEicEYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB be two matrices such that the number of columns of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= equals the number of rows of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Then the product LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZ0aW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJwRidGL0YyRjk= matrix, 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 .Equivalently, the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn column of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is given by the matrix times vector 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 .This stencil might 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4) a) Can you compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn for the matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY5 in exercise 3?b) Let 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. Using 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 compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQUNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQ0FGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn . Huh?Properties for the algebra of matrix addition and multiplication :\302\267 Multiplication is not commutative in general (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn usually does not equal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQkFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn , even if you're multiplying square matrices so that at least the product matrices are the same size).But other properties you're used to do hold:\302\267 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiK0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZDRi8= is commutative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJCRidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GVkZPRjVGK0Y5\302\267 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiK0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZDRi8= is associative 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\302\267 scalar multiplication distributes over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiK0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZDRi8=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 .\302\267 multiplication is associative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSNBQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRidGOi1GMTYlUSJDRidGNEY3LUkjbW9HRiQ2LVEiPUYnRjovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GMTYlUSJBRidGNEY3LUYsNiQtRiM2JC1GMTYlUSNCQ0YnRjRGN0Y6RjpGOg== .\302\267 matrix multiplication distributes over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiK0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZDRi8=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; 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Exercise 5: a) Verify these properties - except for the associative property for multiplication they're all easy to understand. b) For the multiplicative associative property verify that at least the dimensions of the triple product matrices are the same. c) Then check that for the matrices in exercises 3-4, it is indeed true that 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 matrices: The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZ0aW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEYrRjk= identity matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm5GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlRGOkYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB has one's down the diagonal (by which we mean the diagonal from the upper left to lower right corner), and zeroes elsewhere. For example,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 , 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, 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 , etc.In other words, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmZW50cnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Jy1GLzYlUSJpRidGMkY1LUkjbW9HRiQ2LVEifkYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURjpGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlLUYvNiVRIklGJ0YyRjUtRiM2Jy1GLzYlUSJuRidGMkY1LUY+Ni1RKCZ0aW1lcztGJ0ZBRkNGRkZIRkpGTEZORlAvRlNRLDAuMjIyMjIyMmVtRicvRlZGZ29GYG9GMkY1RldGQUZBLUY+Ni1RIj1GJ0ZBRkNGRkZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXXAtSSNtbkdGJDYkUSIxRidGQUZB and 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 if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJSZuZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJqRidGL0YyRjk= .Exercise 6) Check 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, 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 .Hint: for the first equality show that the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn columns of each side agree. For the second equality compare the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn rows.(That's why these matrices are called identity matrices - they are the matrix version of multiplicative identities, e.g. like multiplying by the number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv in the real number system.)On Friday we will continue our discussion of matrix algebra, focusing on:Matrix inverses: A square matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm5GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlRGOkYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB is invertible if there is a matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm5GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlRGOkYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB so thatLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEjQkFGJ0YvRjJGNS1GLDYlUSJJRidGL0YyRjk= .In this case we call LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the inverse of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and write 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 .Remark: A matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= can have at most one inverse, because ifLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEjQkFGJ0YvRjJGNS1GLDYlUSJJRidGL0YyRjk= and also LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjQUNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEjQ0FGJ0YvRjJGNS1GLDYlUSJJRidGL0YyRjk=thenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSNCQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRidGOi1GMTYlUSJDRidGNEY3LUkjbW9HRiQ2LVEiPUYnRjovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GMTYlUSNJQ0YnRjRGN0Y/RjxGOg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRI0FDRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiPUYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSNCSUYnRi9GMkY/RitGPQ==so LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJDRidGL0YyLUY2Ni1RIi5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVkY5Exercise 7a) Verify that for 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 the inverse matrix is 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 .Inverse matrices are very useful in solving algebra problems. For exampleTheorem: If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= exists then the only solution to 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 is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI9RicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSVtc3VwR0YkNiUtRiw2JVEiQUYnRjIvRjdRJ2l0YWxpY0YnLUYjNiYtRj02LVEqJnVtaW51czA7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRl5vLUkjbW5HRiQ2JFEiMUYnRkBGMkZmbi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRiw2KFEiYkYnRi9GMkY0RjZGOUZA .Exercise 7b) Use the theorem and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= in 7a), to write down the solution to the 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Exercise 8) Use matrix algebra to verify why the Theorem is true. Notice that the correct formula is 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 and not 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 (this second product can't even be computed!).