MAT2705-02/06 10F Quiz 5 Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always simplify expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC).1.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
a) Write down the coefficient matrix A, the RHS matrix b and the augmented matrix C = <A | b> for this linear system of equations.b) With technology (identify your choice!), reduce this matrix C step by step to its ReducedRowEchelonForm, recording the intermediate matrices and row operations for each step (as in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RMSZMZWZ0UmlnaHRBcnJvdztGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRiw2JUYuLUYjNiUtRjs2JFEiMkYnRj5GMkY1RkAtRkQ2LVEiLEYnRj5GRy9GS0Y0L0ZNRklGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWlEsMC4zMzMzMzMzZW1GJy1GLDYlRi4tRiM2JC1GOzYkUSIzRidGPkY+RkAtRkQ2LVEoJnNyYXJyO0YnRj5GR0ZKRmBvRk5GUEZSRlRGYW8vRlpGYm9GZW8tRkQ2LVEiK0YnRj5GR0ZKRmBvRk5GUEZSRlQvRldRLDAuMjIyMjIyMmVtRicvRlpGZHBGaW4tRkQ2LVEifkYnRj5GR0ZKRmBvRk5GUEZSRlRGYW9GX3AtRiw2JUYuLUYjNiRGOkY+RkBGXG9GPg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSgmc3JhcnI7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZYLUkmbWZyYWNHRiQ2KEY6LUYjNiQtRjs2JFEiMkYnRj5GPi8lLmxpbmV0aGlja25lc3NHRj0vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYW8vJSliZXZlbGxlZEdGSS1GRDYtUSJ+RidGPkZHRkpGTEZORlBGUkZURlZGWUYrLUYvNiNRIUYnRj4=).
c) Write out the equations that correspond to the reduced matrix. Identify the leading variables and the free variables and solve. State your solution in the scalar form: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRjw= = ..., LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0ZARj4= = ..., etc .d) Now state your solution in column matrix ("vector") form x = ... and then re-express it as an arbitrary linear combination of fixed vectors by factoring out the vector of coefficients of each of the free parameters in the solution.
2. 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 a) Write this linear system in the matrix form A x = b.
b) Write down the inverse coefficient matrix using technology or your memory if good enough, but then verify that its product with A is the identity matrix. Show the matrix multiplication steps by hand to prove that you can actually multiply simple matrices.
c) Now solve this matrix equation using the inverse matrix for the column matrix x , and then write out the individual scalar solutions of the original system for each individual variable.d) Check by backsubstitution into the original two equations that your solution is actually a solution.solution1.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1.QyU+SSVlcW5zRzYiPCUvLCgmSSJ4R0YlNiMiIiIiIiMmRio2I0YtIiIkJkYqNiMiIiVGMyIiIS8sKEYpRjBGLiIiJyZGKjYjRjAhIiRGNC8sKkYpIiImRi4iIipGOEY6RjFGM0Y0RiwtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkRiQ8JEYpRi4=a) The coefficient and RHS matrices, and the augmented matrix are therefore:Qyc+SSJBRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMvSSQlaWRHRiUiKkc1LFYiIiIiPkkiYkdGJS1JJDwsPkdGKjYlIiIhRjVGNUYvPkkiQ0dGJS1JJDx8Z3I+R0YqNiRGJEYxb)d)Now we reduce the augmented matrix and backsubstitute solve the reduced system yielding directly the vector or matrix form of the solution:QyQtJiZJKFN0dWRlbnRHSShfc3lzbGliRzYiNiNJLkxpbmVhckFsZ2VicmFHNiQlKnByb3RlY3RlZEdGJzYjSTFMaW5lYXJTb2x2ZVR1dG9yR0YoNiNJIkNHRigiIiI=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 Done!QydJIkNHNiIiIiItSTZSZWR1Y2VkUm93RWNoZWxvbkZvcm1HRiQ2I0kiJUdGJEYlPkkvVmVjdG9yU29sdXRpb25HRiQvLUkkPCw+R0koX3N5c2xpYkdGJDYmJkkieEdGJDYjRiUmRjI2IyIiIyZGMjYjIiIkJkYyNiMiIiUtSTNCYWNrd2FyZFN1YnN0aXR1dGVHRiRGKA==c) 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 are leading variables, 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 are free variables and the scalar form of the solution is:
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while the two reduced equations that correspond to the reduced matrix were:PkklZXFzMkc2IjwlLyIiIUYnLyZJInhHRiQ2IyIiIiwmJkYqNiMiIiQhIiQmRio2IyIiJSEiKS8mRio2IyIiIywmRi5GOUYyRjQ=If we had done this by hand we would have labeled the two parameters in reversed order.d) Now we pull apart the vector form of the solution into an explicit arbitrary linear combination of fixed vectors:
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We would have let 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 working from top to bottom.2.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KiRJIkFHNiIhIiI=Ly0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0knY29sdW1uR0YpNiMvSSQlaWRHRikiKiFvbjM8LUkwZGVsYXlEb3RQcm9kdWN0RzYkRicvSSttb2R1bGVuYW1lR0YpSSxUeXBlc2V0dGluZ0dGJjYlKiRJIkFHRikhIiJJImJHRilJJXRydWVHRic=We really should separately check the two equations but it is equivalent to just multiplying the solution by the coefficient matrix and comparing to the RHS vector:Ly1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nRzYkRiZJKF9zeXNsaWJHRik2JUkiQUdGKS0mSSdWZWN0b3JHRis2I0knY29sdW1uR0YpNiMvSSQlaWRHRikiKl9eM1YiSSV0cnVlR0YmSSJiR0YpLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=TTdSMApJNlJUQUJMRV9TQVZFLzE0MzAxMDUxNlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIy0iJCIlJiUibUc2JCIiIkYpJkYnNiQiIiNGKSZGJzYkIiIkRikmRic2JEYpRiwmRic2JEYsRiwmRic2JEYvRiwmRic2JEYpRi8mRic2JEYsRi8mRic2JEYvRi8mRic2JEYpIiIlJkYnNiRGLEY+JkYnNiRGL0Y+RiU=TTdSMApJNlJUQUJMRV9TQVZFLzE0MzAxMTAyOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIy0iJCIlIiIkIiIjIiImIiInRiYiIiohIiQiIiFGK0YsIiIlRi1GJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0MzAxMTA5MlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiQiJCIiIUYmRiZGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0MzAxMTE1NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIzAiJCImIiIkIiIjIiImIiInRiYiIiohIiQiIiFGK0YsIiIlRi1GLEYsRixGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0MzAxMTE1NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIzAiJCImIiIkIiIjIiImIiInRiYiIiohIiQiIiFGK0YsIiIlRi1GLEYsRixGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0Mjk5NzU4OFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIzAiJCImIiIiIiIhRidGJ0YmRiciIiQhIiNGJyIiKSEiJUYnRidGJ0YnRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE0Mjk5Nzk3MlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiUiJSYlInhHNiMiIiImRic2IyIiIyZGJzYjIiIkJkYnNiMiIiVGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0Mjk5ODQyMFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiUiJSwmJiUkX3QxRzYjIiIjISIkJkYoNiMiIiIhIiksJkYnRipGLCIiJUYnRixGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0Mjk5OTUwOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIkIiIjIiIlIiIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE0MzA2MDgyOFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyIiJiIjNUYlTTdSMApJNlJUQUJMRV9TQVZFLzE0MzAzNDc0MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyEiIiIiJiMiIiNGKCMiIiVGKCMhIiRGKEYlTTdSMApJNlJUQUJMRV9TQVZFLzE3MDg2NzY4MFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyYlInhHNiMiIiImRic2IyIiI0YlTTdSMApJNlJUQUJMRV9TQVZFLzE0MzA4NjAxNlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyIiKCEiJUYlTTdSMApJNVJUQUJMRV9TQVZFLzUyMzM5Njg4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiImIiM1RiU=TTdSMApJNlJUQUJMRV9TQVZFLzE0MzA2MDgyOFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyIiJiIjNUYl