1.5 Elementary Matrices - Brandon Barba Vaz MAT 343 ONLINE B Spring 2016 Assignment Section 1.5 Elementary Matrices due at 11:59pm MST Solution a E1 is

1.5 Elementary Matrices - Brandon Barba Vaz MAT 343 ONLINE...

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1. (1 point) Suppose that: A = 1 - 4 - 4 4 and B = - 2 5 - 3 4 - 1 1 2 2 5 Given the following descriptions, determine the following ele- mentary matrices and their inverses. a. The elementary matrix E 1 multiplies the first row of A by 1/2. E 1 = , E - 1 1 = b. The elementary matrix E 2 multiplies the second row of A by -2. E 2 = , E - 1 2 = c. The elementary matrix E 3 switches the first and second rows of A. E 3 = , E - 1 3 = d. The elementary matrix E 4 adds 5 times the first row of A to the second row of A. E 4 = , E - 1 4 = e. The elementary matrix E 5 multiplies the second row of B by 1/5. E 5 = , E - 1 5 = f. The elementary matrix E 6 multiplies the third row of B by -2. E 6 = , E - 1 6 = g. The elementary matrix E 7 switches the first and third rows of B. E 7 = , E - 1 7 = h. The elementary matrix E 8 adds 3 times the third row of B to the second row of B. E 8 = , E - 1 8 = Solution: a. E 1 is obtained by multiplying by 1 2 the first row of the identity matrix I = 1 0 0 1 . This gives E 1 = 1 2 0 0 1 The inverse of E 1 is obtained by multiplying the first row of the identity by 2. This gives E - 1 1 = 2 0 0 1 . b. E 2 is obtained by multiplying by - 2 the second row of the identity matrix I = 1 0 0 1 . This gives E 2 = 1 0 0 - 2 The inverse of E 2 is obtained by dividing the first row of the identity by - 2. This gives E - 1 2 = 1 0 0 1 - 2 . c. E 3 is obtained by switching the first and second rows of the identity matrix I = 1 0 0 1 . This gives E 3 = 0 1 1 0 The inverse of E 3 is obtained by switching the first and second rows of the identity . This gives E - 1 3 = 0 1 1 0 . Note that E 3 = E - 1 3 . This is always the case for permutation matrices, that is, matrices that are obtained by permuting the rows of the identity matrix. d. E 4 is obtained by performing the row operation r 2 r 2 + 5 r 1 on the identity matrix I = 1 0 0 1 . This gives E 4 = 1 0 5 1 The inverse of E 4 is obtained by performing the row op- eration r 2 r 2 - 5 r 1 on the identity matrix. This gives E - 1 4 = 1 0 - 5 1 . e. E 5 is obtained by multiplying by 1 5 the second row of the identity matrix I = 1 0 0 0 1 0 0 0 1 . This gives E 5 = 1 0 0 0 1 5 0 0 0 1 The inverse of E 5 is obtained by multiplying the second row of the identity by 5. This gives E - 1 5 = 1 0 0 0 5 0 0 0 1 . f. E 6 is obtained by multiplying by - 2 the third row of the identity matrix I = 1 0 0 0 1 0 0 0 1 . This gives E 6 = 1 0 0 0 1 0 0 0 - 2 The inverse of E 6 is obtained by dividing the third row of the identity by - 2. This gives E - 1 6 = 1 0 0 0 1 0 0 0 1 - 2 . g. E 7 is obtained by switching the fist and the third row of the 1
identity matrix I = 1 0 0 0 1 0 0 0 1 . This gives E 7 = 0 0 1 0 1 0 1 0 0 The inverse of E 7 is obtained by switching the first and third row of the identity . This gives E - 1 7 = 0 0 1 0 1 0 1 0 0 .

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