MATH102-11B Intro to Algebra: Problem Sheet 6.
Assigned : 1 (B C Q) 2 (B C Q) 3 Hand in : Monday 13 September
If A=
a b c d e f g h i
then |A|=aei+bf g+cdh−af h−bdi−ceg
1* Find the determinants of the following matrices [* for B C and Q only]
A =
0 3 −1 0 2 −1
1 1 1
B =
0 1 2 1 0 2 2 2 2
C =
1 2 0 3 1 1 2 1 4
P =
0 0 5
0 2 −7
3 1 3
Q=
1 1 1 1 1 1 1 1 1
R=
0 1 0 2 0 3 4 5 6
2* Find the inverses of the matrices in question 1 (where these exist) using row reduction. [* for B and C and Q only]
3* Verify that the two matrices below are inverses of each other.
X =
1 0 1 1 2 0 2 1 0 1 1 1 1 1 0 1
, Y = 1
2
−1 1 −1 1
−3 1 1 1
−1 1 1 −1
4 −2 0 0
Hence solve −x +y −z +w = 1
−3x +y +z +w = 4
−x +y +z −w =−1
4x −2y = 2
4. If A and B are invertible matrices, show that their productAB is invertible and that (AB)−1 =B−1A−1. Give an example to show that the sum of two invertible matrices need not be invertible.
5. Find the inverse of the following 4×4 matrix. Check your answer by multiplication.
X =
0 1 0 1 1 1 1 1 0 2 0 3 1 1 2 2
6 Give 3×3 elementary matrices which perform the following row operations when multiplied from the left. Also describe the inverse operation and give the ele- mentary matrix for that. Finally find the determinant of each matrix.
a) R3←R3−2R1 b) R2←R2+R3 c) R1← −2R1 d) R2↔R3
7. If a matrix can be written as a product of elementary matrices show that its determinant cannot be zero. Which matrices from question 1 cannot be written as a product of elementary matries?
8. Express the following matrices as products of elementary matrices
D= 1 1 1 2
!
E = 0 3 2 1
!
F =
0 1 1 1 1 1 1 1 2
9. Find the determinants of the elementary matrices used in question 8. Hence find the determinants of the original matrices. Verify that your result agrees with the formula.
10. Use the fact that|AB|=|A| |B|for all n×n matrices A and B to prove that
a) |A−1|= 1
|A| for invertibleA.
b) If|A|= 0 then A cannot be invertible.
c) If |A| 6= 0 then A is invertible.
(Hint - consider the processes involved in finding the inverse and in finding the determinant. Try to show that the former fails precisely when the latter gives an answer of 0.)
