prac final 1

Question 1

how to write the parametric equations of 3 given values that have 3 elements each

Answer 1

literally in the order that they give in the question list them out vertically like in 3 coullms then in the front of each row just do x= , y= , z= and for the first value you leave as is but second value put a s as the variable and third have a t as the variable

Question 2

what makes a matrix singular

Answer 2

if the determinate is zero

Question 3

how to find a value in a matrix that would make it singular

Answer 3

take the determinate and then solve for that variable be careful with the signs

Question 4

whats a example of a matrix with all positive values and a eigen value of 0

Answer 4

basicially the rows have to be a scaler multiple of each other remeber the below answer
(1,1)
(2,2)

Question 5

how to find the cross product

Answer 5

bascially set up a 3x3 matrix with the top row being i,j,k and then just follow the steps for the determinate but then just solve the stuff in the brackets this should give 3 values and but them in brackets thats it done

Question 6

question where they show a bunch of row operations and give the new matrix and ask for the determinate of the OG matrix

Answer 6

• take the determinate of the new matrix
• if you swap any rows you change the sign of the determinate
• if you multiply do the inverse of the multiplication eg( x 3 = 1/3)

Question 7

4x4 matrix determinate

Answer 7

say its 0 cuz its upper triangular or smt

Question 8

3x3 matrix given 1 eigen value prove that thats a value and find the other value

Answer 8

the bottom right of the matrix is a 1 just remove that to get a 2x2 matrix and with that easily do it youll get 1 and 3 so that proves one and finds another vlaue

Question 9

Let (b) W be the eigenspace of B corresponding to λ = 1. Find a basis for W

Answer 9

take the original matrix minus the standard matrix and the matrix u get from that make the colums the vector basis for a

Question 10

A linear operator (a) T is the composition of the following linear operators in R2 find the standard matrix
( scaling, rotation, reflection) find the standard matrix

Answer 10

• scaled the standard matrix by x factor 2 and y factor by 3
• rotation matrix is 0 , -1 , 1, 0
• reflection about the y axis you just switch the columns of the rotation matrix

then in the reverse order listed in the question take the dot product of the three matricies
- reflection x rotation x scaling this gets teh standard matrix

Question 11

Determine the line of intersection of the two planes. Write the line in vector form.

Answer 11

make the z variable 0 and then solve for the other variables

put the variables in (x , y, z)

take the norm of both equations then take the cross product of the norms having the first equation being row 2
- solve the values the brackets of the cross pord and leave as is giving 3 values

input into the following equation
r = postion vector(1st one) + T(2nd one)

Question 12

how to Determine the cosine of the angle formed by the two planes.

Answer 12

dot product of the 2 norms over the norm of the norms times each other

Question 13

what to do if its asking for the sin angle

Answer 13

square root 1- cos^2

Question 14

how to find the exponet of a matrix that is diagnol ( specifically uper tri)

Answer 14

what ever is the exponent you times that by the value thats not 1 or zero

Question 15

how to find the negative cube of a matrix

Answer 15

take the inverse of the matrix and then take the dot prod of the same matrix