Name _______Answers_________       Linear Algebra, Quiz 1, Summer 2004         Page 1

 

1.  Find the solution to the following system of equations by row reducing the augmented matrix, and check your answer.

`Problem 1.  System of Equations:`

-x+3*y-4*z = -4

2*x+4*y+z = 9

-4*x+2*y-z = -1

`Problem 1.  Matrix Equation:`

A*X = B

matrix([[-1, 3, -4], [2, 4, 1], [-4, 2, -1]])*matrix([[x], [y], [z]]) = matrix([[-4], [9], [-1]])

`Problem 1.  Augmented Matrix:`

`Augment A:  `*`A'` = matrix([[-1, 3, -4, -4], [2, 4, 1, 9], [-4, 2, -1, -1]])

`Row reduce using (-1): `*matrix([[-1, 3, -4, -4], [2, 4, 1, 9], [-4, 2, -1, -1]])*` -->> `*matrix([[1, -3, 4, 4], [2, 4, 1, 9], [-4, 2, -1, -1]])

`Row reduce using (4),(-2): `*matrix([[1, -3, 4, 4], [2, 4, 1, 9], [-4, 2, -1, -1]])*` -->> `*matrix([[1, -3, 4, 4], [0, 10, -7, 1], [0, -10, 15, 15]])

`Swap row2 & row3: `*matrix([[1, -3, 4, 4], [0, 10, -7, 1], [0, -10, 15, 15]])*` -->> `*matrix([[1, -3, 4, 4], [0, -10, 15, 15], [0, 10, -7, 1]])

`Row reduce using (-1/10): `*matrix([[1, -3, 4, 4], [0, -10, 15, 15], [0, 10, -7, 1]])*` -->> `*matrix([[1, -3, 4, 4], [0, 1, -3/2, -3/2], [0, 10, -7, 1]])

`Row reduce using (-10),(3): `*matrix([[1, -3, 4, 4], [0, 1, -3/2, -3/2], [0, 10, -7, 1]])*` -->> `*matrix([[1, 0, -1/2, -1/2], [0, 1, -3/2, -3/2], [0, 0, 8, 16]])

`Row reduce using (1/8): `*matrix([[1, 0, -1/2, -1/2], [0, 1, -3/2, -3/2], [0, 0, 8, 16]])*` -->> `*matrix([[1, 0, -1/2, -1/2], [0, 1, -3/2, -3/2], [0, 0, 1, 2]])

`Row reduce using (3/2),(1/2): `*matrix([[1, 0, -1/2, -1/2], [0, 1, -3/2, -3/2], [0, 0, 1, 2]])*` -->> `*matrix([[1, 0, 0, 1/2], [0, 1, 0, 3/2], [0, 0, 1, 2]])

`Row Reduction:  `*matrix([[-1, 3, -4, -4], [2, 4, 1, 9], [-4, 2, -1, -1]])*` -->>  ...  -->>`*matrix([[1, 0, 0, 1/2], [0, 1, 0, 3/2], [0, 0, 1, 2]])

`Solution Vector:  `*matrix([[x], [y], [z]]) = matrix([[1/2], [3/2], [2]])

`Check Matrix Equation: `*matrix([[-1, 3, -4], [2, 4, 1], [-4, 2, -1]])*matrix([[1/2], [3/2], [2]]) = matrix([[-4], [9], [-1]])

`Check Matrix Equation: `*matrix([[-4], [9], [-1]]) = matrix([[-4], [9], [-1]])

2.  Find the elementary matrices corresponding to the first six elementary row operations you used in 1.  above, and then multiply them together as follows to find the matrix F:

E[6]*E[5]*E[4]*E[3]*E[2]*E[1] = F

`Problem 2.  Elementary Matrices`

A = matrix([[-1, 3, -4], [2, 4, 1], [-4, 2, -1]])

`Elementary Row Operation 1`, proc (Row1) options operator, arrow; -Row1 end proc

`Elementary Matrix 1:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[1] = matrix([[-1, 0, 0], [0, 1, 0], [0, 0, 1]]), E[1]^`-1` = matrix([[-1, 0, 0], [0, 1, 0], [0, 0, 1]])

``

`Elementary Row Operation 2`, proc (Row2) options operator, arrow; Row2-2*Row1 end proc

`Elementary Matrix 2:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[2] = matrix([[1, 0, 0], [-2, 1, 0], [0, 0, 1]]), E[2]^`-1` = matrix([[1, 0, 0], [2, 1, 0], [0, 0, 1]])

``

`Elementary Row Operation 3`, proc (Row3) options operator, arrow; Row3+4*Row1 end proc

`Elementary Matrix 3:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[3] = matrix([[1, 0, 0], [0, 1, 0], [4, 0, 1]]), E[3]^`-1` = matrix([[1, 0, 0], [0, 1, 0], [-4, 0, 1]])

``

`Elementary Row Operation 4`, `Swap Row2 and Row3`

`Elementary Matrix 4:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[4] = matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]]), E[4]^`-1` = matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])

``

`Elementary Row Operation 5`, proc (Row2) options operator, arrow; -1/10*Row2 end proc``

`Elementary Matrix 5:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[5] = matrix([[1, 0, 0], [0, -1/10, 0], [0, 0, 1]]), E[5]^`-1` = matrix([[1, 0, 0], [0, -10, 0], [0, 0, 1]])

``

`Elementary Row Operation 6`, proc (Row1) options operator, arrow; Row1+3*Row2 end proc

`Elementary Matrix 6:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[6] = matrix([[1, 3, 0], [0, 1, 0], [0, 0, 1]]), E[6]^`-1` = matrix([[1, -3, 0], [0, 1, 0], [0, 0, 1]])

``

`Elementary Row Operation 7`, proc (Row3) options operator, arrow; Row3-10*Row2 end proc

`Elementary Matrix 7:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[7] = matrix([[1, 0, 0], [0, 1, 0], [0, -10, 1]]), E[7]^`-1` = matrix([[1, 0, 0], [0, 1, 0], [0, 10, 1]])

``

`Elementary Row Operation 8`, proc (Row3) options operator, arrow; 1/8*Row3 end proc

`Elementary Matrix 8:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[8] = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1/8]]), E[8]^`-1` = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 8]])

``

`Elementary Row Operation 9`, proc (Row1) options operator, arrow; Row1+1/2*Row3 end proc

`Elementary Matrix 9:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[9] = matrix([[1, 0, 1/2], [0, 1, 0], [0, 0, 1]]), E[9]^`-1` = matrix([[1, 0, -1/2], [0, 1, 0], [0, 0, 1]])

``

`Elementary Row Operation 10`, proc (Row2) options operator, arrow; Row2+3/2*Row3 end proc

`Elementary Matrix 10:  `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])*` -->> `*E[10] = matrix([[1, 0, 0], [0, 1, 3/2], [0, 0, 1]]), E[10]^`-1` = matrix([[1, 0, 0], [0, 1, -3/2], [0, 0, 1]])

``

E[2]*E[1]*`=`*matrix([[1, 0, 0], [-2, 1, 0], [0, 0, 1]])*matrix([[-1, 0, 0], [0, 1, 0], [0, 0, 1]]) = matrix([[-1, 0, 0], [2, 1, 0], [0, 0, 1]])

E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, 1, 0], [4, 0, 1]])*matrix([[-1, 0, 0], [2, 1, 0], [0, 0, 1]]) = matrix([[-1, 0, 0], [2, 1, 0], [-4, 0, 1]])

E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])*matrix([[-1, 0, 0], [2, 1, 0], [-4, 0, 1]]) = matrix([[-1, 0, 0], [-4, 0, 1], [2, 1, 0]])

E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, -1/10, 0], [0, 0, 1]])*matrix([[-1, 0, 0], [-4, 0, 1], [2, 1, 0]]) = matrix([[-1, 0, 0], [2/5, 0, -1/10], [2, 1, 0]])

E[6]*E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 3, 0], [0, 1, 0], [0, 0, 1]])*matrix([[-1, 0, 0], [2/5, 0, -1/10], [2, 1, 0]]) = matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [2, 1, 0]])

3. Find the six inverses of the six elementary matrices you found in 2. above, and multiply these together in such a way that you get the inverse of F. Check that you found the inverse of F by multiply your answer times the matrix F.

E[6]*E[5]*E[4]*E[3]*E[2]*E[1] = F

(E[6]*E[5]*E[4]*E[3]*E[2]*E[1])^`-1` = F^`-1`

E[1]^`-1`*E[2]^`-1`*E[3]^`-1`*E[4]^`-1`*E[5]^`-1`*E[6]^`-1` = F^`-1`

matrix([[-1, 0, 0], [0, 1, 0], [0, 0, 1]])*matrix([[1, 0, 0], [2, 1, 0], [0, 0, 1]])*matrix([[1, 0, 0], [0, 1, 0], [-4, 0, 1]])*matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])*matrix([[1, 0, 0], [0, -10, 0],...

E[5]^`-1`*E[6]^`-1`*` = `*matrix([[1, 0, 0], [0, -10, 0], [0, 0, 1]])*matrix([[1, -3, 0], [0, 1, 0], [0, 0, 1]]) = matrix([[1, -3, 0], [0, -10, 0], [0, 0, 1]])

E[4]^`-1`*E[5]^`-1`*E[6]^`-1`*` = `*matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])*matrix([[1, -3, 0], [0, -10, 0], [0, 0, 1]]) = matrix([[1, -3, 0], [0, 0, 1], [0, -10, 0]])

E[3]^`-1`*E[4]^`-1`*E[5]^`-1`*E[6]^`-1`*` = `*matrix([[1, 0, 0], [0, 1, 0], [-4, 0, 1]])*matrix([[1, -3, 0], [0, 0, 1], [0, -10, 0]]) = matrix([[1, -3, 0], [0, 0, 1], [-4, 2, 0]])

E[2]^`-1`*E[3]^`-1`*E[4]^`-1`*E[5]^`-1`*E[6]^`-1`*` = `*matrix([[1, 0, 0], [2, 1, 0], [0, 0, 1]])*matrix([[1, -3, 0], [0, 0, 1], [-4, 2, 0]]) = matrix([[1, -3, 0], [2, -6, 1], [-4, 2, 0]])

E[1]^`-1`*E[2]^`-1`*E[3]^`-1`*E[4]^`-1`*E[5]^`-1`*E[6]^`-1`*` = `*matrix([[-1, 0, 0], [0, 1, 0], [0, 0, 1]])*matrix([[1, -3, 0], [2, -6, 1], [-4, 2, 0]]) = matrix([[-1, 3, 0], [2, -6, 1], [-4, 2, 0]])*...

`Check:  `*F^`-1`*F = matrix([[-1, 3, 0], [2, -6, 1], [-4, 2, 0]])*matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [2, 1, 0]])*` = `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

4. [Extra Credit] This is a continuation of 2. above. Find all the remaining elementary matrices corresponding to the rest of the elementary row operations that you used in 1. Use these together with those found in 2. to compute the matrix inverse of A, and check that it is the inverse of A. Note that your answer must be correct to receive this extra credit.

`Now that we have all the Elementary Matrices we can compute the inverse of A:`

E[10]*E[9]*E[8]*E[7]*E[6]*E[5]*E[4]*E[3]*E[2]*E[1] = A^`-1`*`=`*matrix([[3/40, 1/16, -19/80], [1/40, 3/16, 7/80], [-1/4, 1/8, 1/8]])

``

E[2]*E[1]*`=`*matrix([[1, 0, 0], [-2, 1, 0], [0, 0, 1]])*matrix([[-1, 0, 0], [0, 1, 0], [0, 0, 1]]) = matrix([[-1, 0, 0], [2, 1, 0], [0, 0, 1]])

E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, 1, 0], [4, 0, 1]])*matrix([[-1, 0, 0], [2, 1, 0], [0, 0, 1]]) = matrix([[-1, 0, 0], [2, 1, 0], [-4, 0, 1]])

E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])*matrix([[-1, 0, 0], [2, 1, 0], [-4, 0, 1]]) = matrix([[-1, 0, 0], [-4, 0, 1], [2, 1, 0]])

E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, -1/10, 0], [0, 0, 1]])*matrix([[-1, 0, 0], [-4, 0, 1], [2, 1, 0]]) = matrix([[-1, 0, 0], [2/5, 0, -1/10], [2, 1, 0]])

E[6]*E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 3, 0], [0, 1, 0], [0, 0, 1]])*matrix([[-1, 0, 0], [2/5, 0, -1/10], [2, 1, 0]]) = matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [2, 1, 0]])

E[7]*E[6]*E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, 1, 0], [0, -10, 1]])*matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [2, 1, 0]]) = matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [-2, 1, 1]])

E[8]*E[7]*E[6]*E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1/8]])*matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [-2, 1, 1]]) = matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [-1/4, 1/8, 1/...

E[9]*E[8]*E[7]*E[6]*E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 1/2], [0, 1, 0], [0, 0, 1]])*matrix([[1/5, 0, -3/10], [2/5, 0, -1/10], [-1/4, 1/8, 1/8]]) = matrix([[3/40, 1/16, -19/80], [2/5, 0, -1/10]...

E[10]*E[9]*E[8]*E[7]*E[6]*E[5]*E[4]*E[3]*E[2]*E[1]*`=`*matrix([[1, 0, 0], [0, 1, 3/2], [0, 0, 1]])*matrix([[3/40, 1/16, -19/80], [2/5, 0, -1/10], [-1/4, 1/8, 1/8]]) = matrix([[3/40, 1/16, -19/80], [1/4...

`Check:  `*A^`-1`*A = matrix([[3/40, 1/16, -19/80], [1/40, 3/16, 7/80], [-1/4, 1/8, 1/8]])*matrix([[-1, 3, -4], [2, 4, 1], [-4, 2, -1]])*` = `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

`Check:  `*A^`-1`*A = 1/80*matrix([[6, 5, -19], [2, 15, 7], [-20, 10, 10]])*matrix([[-1, 3, -4], [2, 4, 1], [-4, 2, -1]])*` = `*matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])