1. Input the matrix 12 12 12 12 12 17 7 7 7 12 12 12 12 12 7 1 7 7 7 12 12 12 12
ID: 3114299 • Letter: 1
Question
1. Input the matrix 12 12 12 12 12 17 7 7 7 12 12 12 12 12 7 1 7 7 7 12 12 12 12 12 7 7 1 7 7 12 12 12 12 12 7 7 7 1 7 12 12 12 12 12 7 7 771] Note: In the real test, this matriz would be provided in an m-file. You would not have to enter it manually (a) Calculate the dot product of the 3rd row and 6th column of P. (b) Let Q = 2p-1. Calculate det PQ. (c) Calculate rank (P+ 6/). (d) Let a denote the vector corresponding to the first column of P, let b denote the vector corresponding to the sixth column, and let c denote the vector corresponding to the third row. Calculate the fourth entry of the linear combination -a+ 2b-2cExplanation / Answer
(a) The dot product of the 3rd row and the 6th column of P is (0,0,4,0,0,7,7,7,7,7).(7,7,7,7,7,1,7,7,7,7) = 0+0+4*7+0+0+7*1+7*7+7*7+7*7+7*7 = 28+7+49+49+49+49= 231.
(b) If Q = 2P-1, then PQ = P(2P-1) = 2PP-1 = 2I10 =
2
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(c ) P+6I =
10
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The last 5 rows od P+6I are identical. Therefore, the the rank of this matrix will be 6. On dividing the first 5 rows by 10, and on adding -1 time the 6th row to each of the 7th,8th, 9th, and 10th row, we will get a matrix with 6 non-zero rows. Then, on adding -12 time each of 1st,2nd,3rd,4th,5th rows to the 6th row and then on multiplying the 6th row by 1/7, we can onserve that the RREF of this matrix is
1
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It has 6 non-zero rows. Therefore, the rank of P+6I is 6.
(d) –a+2b-2c = -(4,0,0,0,0,12,12,12,12,12)T +2(7,7,7,7,7,1,7,7,7,7)T-2(0,0,4,0,0,7,7,7,7,7)T = (-4, 0,0,0,0,-12,-12,-12,-12,-12)T+(14,14,14,14,14,2,14,14,14,14)+(0,0,-8,0,0,-14,-14,-14,-14,-14)T = ( 10,14,6,14,14,-24,-12,-12,-12,-12)T. The 4th entry of the linear combination –a+2b-2c is 14.
2
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2
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