please help me with my homework and please answer it by handwritten Find the vec

Question

please help me with my homework and please answer it by handwritten

Find the vector found by rotating vector [-1/ ] on the xy-plane counter clockwise for 330 degree about the origin via a matrix-vector multiplication. Consider matrix A = . Apply the Gaussian elimination algorithm to find the reduced row echelon form R of matrix A. Specify the elementary row operation performed in each step. Use definition to determine the rank and nullity of matrix A. {Remark: In case you have difficulty in the procedure of Gaussian elimination, you may use tools to find R and then continue to answer part (b).} A commuter airline wants to purchase a fleet of 30 airplanes with a combined carrying capacity of 840 passengers. The three available types of planes carry 16, 24, and 40 passengers, respectively. Determine all possible purchase options. Find how many planes of each type to be purchased if the number of 40-passengers planes purchased is the least.

Explanation / Answer

3)

let A be the plane which can carry 16 passengers

let B be the plane which can carry 24 passengers

let C be the plane which can carry 40 passengers

A+B+C=30 ------------ (1)

16A+24B+40C=840

2A+3B+5C=105 ------------- (2)

(A,B,C) = (1,21,8),(3,18,9),(5,15,10),(7,12,11),(9,9,12),(11,6,13),(13,3,14) are the possible values

least value of 40 passenger flights is 8 and the values of (A,B,C) = (1,21,8)

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