MATH 240 QUIZ 3 NAME SOLUTIONS
Prof. J. Beachy Friday, 9/14/2007 Score / 20
1. (6 pts) Find the reduced row echelon form of the matrix A=cos θsin θ
−sin θcos θ.
Remember that a b
c d is invertible precisely when the determinant ad −bc is nonzero. The determinant of the
given matrix is cos2θ+ sin2θ= 1, so the matrix is invertible, and then our theorems guarantee that it can be row
reduced to the identity matrix.
2. (7 pts) The matrix A=1 2
3 4 is invertible. Find a sequence of elementary matrices that will reduce Ato the
identity.
1 0
−3 1 1 2
3 4 =1 2
0−2
1 0
0−
1
2 1 0
−3 1 1 2
3 4 =1 0
0−
1
2 1 2
0−2=1 2
0 1
1−2
0 1 1 0
0−
1
2 1 0
−3 1 1 2
3 4 =1−2
0 1 1 2
0 1 =1 0
0 1
E1=1 0
−3 1 E2=1 0
0−
1
2E3=1−2
0 1
This isn’t part of the question, but we can use this sequence to write Aas a product of elementary matrices.
1 2
3 4 =1 0
−3 1 −11 0
0−
1
2−11−2
0 1 −1
=1 0
3 1 1 0
0−2 1 2
0 1
3. (7 pts) Find the inverse of the following matrix (if it exists): A=
111
123
011
111100
123010
011001
111 100
012−110
011 001
111 100
011 001
012−110
100 10−1
0 1 1 0 0 1
001−1 1 −1
1 0 0 1 0 −1
010 1−1 2
001−1 1 −1
111
123
011
−1
=
1 0 −1
1−1 2
−1 1 −1
