3.6.6. Read section 3.4 and then try the following: (a) Find the matrix E which swaps...

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3.6.6. Read section 3.4 and then try the following: (a) Find the matrix E which swaps the first and second rows of a 4 x 7 matrix. Then show this matrix is invertible by multiplying it by itself. (Why does this make sense?) (b) Find the matrix E' which scaled the fourth row of a 4 x 7 matrix by a = 10. What about by a factor of a' = 1. Then show these matrices are invertible by multiplying them. (c) Find the matrix E" which performs row2 row2-7row. What new row operation "undoes" this operation? What is the corresponding matrix? Show me that these two matrices are inverse to one another. (d) Show me the two ways that you can perform the three elementary row operations I asked you you to consider in each part above (swap first and second, scale the fourth row by a factor, and then the first combination): both by actually performing the operations by hand and then by multiplying the corresponding matrices E" E E.A. (e) Now since E" E' E A is the resulting matrix from the reduction algorithm, it is some upper triangular matrix, U, so we have A E" E E A=U. Find the inverse for E", and multiple both sides on the left (order matters!) to obtain, E"E" E' E A E' E A=E"-U. Now repeat the same for E' and E, and finally multiply these inverse matrices together to find L such that A = LU. Congratulations, you have worked out how row reduction is also a matrix factoring algorith 3.6.6. Read section 3.4 and then try the following: (a) Find the matrix E which swaps the first and second rows of a 4 x 7 matrix. Then show this matrix is invertible by multiplying it by itself. (Why does this make sense?) (b) Find the matrix E' which scaled the fourth row of a 4 x 7 matrix by a = 10. What about by a factor of a' = 1. Then show these matrices are invertible by multiplying them. (c) Find the matrix E" which performs row2 row2-7row. What new row operation "undoes" this operation? What is the corresponding matrix? Show me that these two matrices are inverse to one another. (d) Show me the two ways that you can perform the three elementary row operations I asked you you to consider in each part above (swap first and second, scale the fourth row by a factor, and then the first combination): both by actually performing the operations by hand and then by multiplying the corresponding matrices E" E E.A. (e) Now since E" E' E A is the resulting matrix from the reduction algorithm, it is some upper triangular matrix, U, so we have A E" E E A=U. Find the inverse for E", and multiple both sides on the left (order matters!) to obtain, E"E" E' E A E' E A=E"-U. Now repeat the same for E' and E, and finally multiply these inverse matrices together to find L such that A = LU. Congratulations, you have worked out how row reduction is also a matrix factoring algorith