Linear Algebra: Week 11 - 15 Week

x^2+3x>10

(-infinity,-5)U(2,infinity)

(x+2)/(x-1)_<3

(-infinity,1)U[5/2, infinity)

Find the determinant of this triangular matrix [2 0 0] [4 -2 0] [1 1 1]

(2)(-2)(1) = -4

[5x+y=9 [2x-4y=14

(25/11, -26/11)

Solve the system. X=A^-1B X-y=6 x-z=4 6x-2y-3z=2 [1 -1 0] [1 0 -1] [6 -2 -3]

(x,y,z) = (-22, -28, -26)

Solve using Gauss elimination and back substitution [x-2y+3z = 9 [ -x+3y+z=-2 [2x-5y+5z=17

(x,y,z) = (1,-1,2)

Solve for X in the equation 3X + A = B A = B= (0 1) (-3 4) (2 -3) (3 5)

-1 1 1/3 8/3

Find the inverse of [1 -1 0] [1 0 -1] [6 -2 -3]

-2 -3 1 -3 -3 1 -2 -4 1

Find (-1) (2) (0) - (5) (2) (3)

-3 -5 -1

Use the above formula to find the inverse of A [1 4] [-1 -3]

-3 -4 1 1

Find the determinant of the following matrix: [-5 0 -1] [1 2 -1] [-3 4 1]

-40

What number must the first entry of each row be?

1

Find AB if A= B= (1 0) (1 2 3) (0 1) (4 5 6)

1 2 3 4 5 6

If the coefficient matrix A of a square system is invertible, how many unique solutions does it have?

1 unique solution

List the 4 steps to matrix multiplication

1. 1st Row x 1st Column 2. 1st Row x 2nd Column 3. 2nd Row x 1st Column 4. 2nd Row x 2nd Column

Describe the 5 steps to decode a secret messege

1. Calculate A^-1 from the given matrix A using the inverse formula 2. Rewrite your cryptogram into 1x2 coded matrices 3. Multiply each coded matrix by A^-1, forming un-coded matrices 4. Write your sequence of un-coded matrices in order and use the alphabet/number code given to decode the messege

Describe the the 7 steps to encode a secret message

1. Choose an invertable 2x2 matrix (matrix A) 2. Pick a messege between 10 and 12 characters (including spaces and punctuation) 3. Split your matrix into 1x2 matrices in order of letters and including spaces 4. Form un-coded matrices using an alphabet to number code. 5. Multiply each uncoded matrix by matrix A using matrix multiplication 6. Write your new sequence of coded matrices in order 7. Rewrite your sequence using commas instead of matrix form = cryptogram

Describe the 4 steps of the method of graphing

1. Graph both equations for y in terms of x 2. View both equations in the same viewing window 3. Use the trace feature to find the points of intersection of the graphs 4. Check that your solution satisfies both equations

Describe the 4 steps to solving polynomial inequalities

1. Make sure there is a 0 to the right of the equation 2. Solve the polynomial 3. Write a numberline and determine regions of interest 4. Test points in each region. If they satisfy the equation and the inequality sign's requirements, write the answer to include them and to leave out the regions that do not work

Describe the 5 steps of the method of elimination

1. Manipulate the equations so that one of your coefficients will be eliminated if the equations were added together 2. Add the equations so that you completely eliminate one variable 3. Solve for the remaining variable 4. Plug the answer from step 3 back into one of the origional equations to find the missing variable 5. Check that your solutions atisfy both of the origional equations

Describe the 5 steps of the method of substitution

1. Solve on of the equations for one variable 2. Substitute the expression in step 1 into the second equation 3. Solve the equation for the remaining variable 4. Plug the answer from step 3 into one of the origional equations to find the missing variable 5. Check that your solutions satisfy both equations

Describe the three steps to solving a system of equations using Inverse

1. Write out the system AX=B and mulitply both sides by A^1 to get X=A^-1B 2. Find A^-1 using matrices 3. Solve for X. The answer will be in the form [x] [y] [...]

What are the 4 steps to find the inverse of a square matrix using the identity matrix and Gauss elimination?

1. Write the augmented matrix as [ A | I ] Where A is the nxn matrix (the origional square matrix) and I is the identity matrix. 2. If possible, reduce A to I using elementary row operations on the entire matrix (Including the identity matrix) 3. You will be left with the matrix [ I | A^-1 ] 4. Check your work by verifying AA^-1 = I = A^-1A

Elementary Row Operations (3)

1. interchange 2.Multiply by non-zero constant 3. Add one equation to another

What are the three things you can do when using gaussian elimination?

1. interchange two equations 2. Multiply one the equations by a nonzero constant 3. Add a multiple of one equation to another equation

Write the easier formula for finding the inverse of a 2x2 matrix A = [a b] [c d]

1/(ad-bc) x [d -b] [-c a]

Find th determinant of this matrix: [-1 2 1] [1 0 1 ] [0 1 0]

2

Find 2B-C if B = C = (4 -2 5) (3 6 6) (3 0 4) (0 4 2) (-6 7 -1) (9 -8 1)

5 -10 4 6 -4 6 -21 22 -3

Decode 9, 36, -3, 3, 17, 73, -25, -75, -6, -3, 28, 112 Matrix A= [1 4] [-1 -3]

9, 0, 12, 15, 22, 5, 0, 25, 15, 21, 28, 0 I Love You!

Incode the messege: I Love You! Matrix A= [1 4] [-1 -3]

9, 36, -3, 3, 17, 73, -25, -75, -6, -3, 28, 112

What is an augmented matrix

A matrix that was created from a linear system of equations

What is a determinant and what is the determinant formula?

A real number associated with a square matrix. Formula: ad-bc

Identity matrix

A square matrix with ones (1s) along the main diagonal, from the upper left element to the lower right element, and zeros (0s) everywhere else.

[ x^2+4x-y=7 [ 2x-y=-1

Answers: (-4,-7) and (2,5)

Find the area of a triangle whose vertices are (1,0), (2,2) and (4,3)

Area = 3/2

Fine the area of a parallelogram with vertices: (0,0) (2,0) (1,3) and (3,3)

Area = 6

If a row has onlly zeros where should it be placed?

As the last row

What is B in matrices?

B will be given as a matrix or as the answers to given equations Ex. x+4y=8 -x-3y=2 B= [8] [2]

How do you add/subtract two matrices?

By adding/subtracting their corresponding entries

How do you get row echelon form?

By using gaussian elimination

If the equation has an x^3, what method can you use to eliminate the ^3 exponent if there is no common solution?

Find somthing x can equal through using p/q and then use that to perform synthetic division to get an equation to the second degree. From there find the remaining zeros

How do you graph rational (x+#)/(x-#) on a number line

First, make sure the number on the right side is 0. Second, solve the x in the numerator and graph that number. Then solve for x in the denominator and that shows what x cannot equal. Make sure to graph that one with an open circle.

When is a system of linear equations inconsistent?

If it has no solution

AxA^-1 =

In (Identity matrix, nxn because it is a square)

What does a multiplicity of zero indicate for a number line

Multiplicity of zero means that on a number line, each section will have alternating signs (- or +)

What is scaler multiplication

Multiplying an entire matrix by one variable

[-x+y=4 [x^2+y=3

No solution

What number of solutions can a system of linear equations have exactly (3 types)?

One solution, infinently many solutions, or no solution

If the equation has an x^3, what method can you use to eliminate the ^3 exponent if there is a common factor set?

Separate the equation into two parts, one of which you can factor out an x. Ex. x^3-3x^2-x+3<0 x^2(x-3)-1(x-3)

If the system is dependent (x+y+z=0) what should you do?

Set z=a and solve for x and y in terms of a

If 0<k<1 are you strecthing or shrinking the square?

Shrinking the square

How do you find the determinant of a 3x3 matrix?

Split the matrix into three 2x2 matrices and find the determinant of each x the correlating first row number

If k > 1 are you stretching or shrinking the square?

Stretching

What will the outer dimensions (m in A and n in B) represent in the result?

The dimensions of the result

If you want to perform matrix multiplication what dimensions must be the same?

The inner dimensions Ex. (2x2)(2x3) works because n=2 in A and m=2 in B

Image

The transformed figure

[2x+4y-2z = 0 [3x+y=1

Turn back into an equation once all possible steps are completed and you can see it will have infinently many solutions. x+2y-z=0 y-3z=-1 z=4 x=-5a+1

How do you transform a square with given vertices?

Turn each vertex into a 2x1 matrix. Take the appropriate 2x2 matrix (the ones mentioned above) and multiply each 2x1 matrix by it. Remember you must multiply in the order of (2x2)(2x1)

When do you use an open vs closed circle on a graph?

Us a closed circle when the graph can equal that number and use an open circle when the graph cannot equal that number.

How do you show your steps when solving multivariable linear systems?

Use arows with the operations you are going to do written above to connect each new step

What is row echelon form?

When 1's go down the main diagonal from upper left to lower right with 0's below the 1's in the first two columns

When is a system of linear equations consistant?

When it has at least one solution

How do you determine if two matrices are inverses of eachother?

When multiplied they result in an identity matrix

When does a system of linear equations have infinate solutions

When the equations cancel out completly 0=0

Can you multiply A^-1 x A and still get the identity matrix?

Yes

How do you find the area of a parallelogram with the vertices and what vertices do you need?

You need (0,0), (a,b), (c,d), (a+c, b+d) Area = det|A| (ad-bc) = [a b] [c d]

To refelect the square (given by a matrix) in the y-axis use the matrix....

[-1 0] [0 1]

x^3-2x^2_>5x-6 (use method demonstrated above)

[-2,1]U[3,infinity)

To reflect the square on th x-axis use the matrix....

[1 0] [0 -1]

To vertically stretch or shrink the square use the matrix....

[1 0] [0 k]

Horizontally stretch a square with the following vertices by a factor of 2: (1,3), (3,3), (1,0), (3,0)

[2] [2] [6] [6] [3] [0] [3] [0]

To horizontally stretch or shrink the square use the matrix....

[k 0] [0 1]

triangular matrix

a matrix with all zeros either above or below the main diagonal

What is a column matrix?

a matrix with only one column

What is a row matrix?

a matrix with only one row

What is a square matrix?

a matrix with the same number of rows and columns

What is the distributive property?

a(b+c)=ab+ac

When are two matrices equal?

if they have the same order (mxn) and all their corresponding entries are equal

What is back substitution?

inputting your first answer from the third row into the second row and then both answers into the first row

If a matrix has an inverse it is called________

inverted

What does m and n stand for in an mxn matrix?

m: number of rows n: number of columns

How to graph and shade systems of linear inequalities

make sure there is only one variable on one side and graph the line. If the sign is includes an equal make the line solid and if it is only a > or <, make the line dashed. Then shade the appropriate side of the graph and the area that satisfies both graphs is the solution.

What are the solutions to systems of equations?

ordered pairs

Three points (x1,y1), (x2,y2), (x3,y3) are colliner is and only if________

the determinant = 0

How do you find the determinant of a triangular matrix?

the determinant is the product of the diagonals

Pre-image

the original figure

Find the equation of the line passing through the points (2,4) and (-1,3) using the determinant formula above

y = x/3 + 10/3

How do you find the area of a triangle with the vertices (x1,y1), (x2,y2), (x3,y3)

| x1 y1 1| +- 1/2 |x2 y2 1| |x3 y3 1| (same method as regular 3x3 matrix)

Determine whether the points (-2,-2), (1,1), and (7,5) are collinear

|A| = -6 Non collinear

An equation of the line passing through two distinct points (x1, y1) and (x2,y2) is given by ..... (write the determinant equation)

|x y 1| |x1 y1 1| = 0 |x2 y2 1|