Math 205B&C 03/06/09 Quiz 05 page 1 Name 8am 1:10 pm
1. Suppose A4=
1239−18
034 7
001 6
000 4
, and Ais some 4 ×4 matrix.
Suppose that A∼A1∼A2∼A3∼A4=A4, and the following row operations are applied sequentially
to turn Ainto A4:
Matrix A1: Row 3 of Ais divided by 4.
Matrix A2:InA1, 2 copies of Row 4 are added to Row 2.
Matrix A3:InA2, Row 4 is multiplied by 3.
Matrix A4=A4: Rows 1 and 4 of A3are swapped.
1A. Find det(A4).
1B. Find det(A).
1C. Find and label the original matrix A, and matrices A1,A2and A3.
1D. Suppose those same four operations were applied to A4itself (so the first operation is “Row 3 of
A4is divided by 4”, and so on). Find the determinant of the resulting matrix M(you do not need to
find M).
2. Suppose B=
abc
pqr
xyz
, and det(B) = 3. Find each of the following:
2A. det(B·B)2B. det(B5)
2C.
5p5q5r
abc
x−7ay−7bz−7c
2D. det(5B)
2E. det(B+B)2F. det(BT)2G. det(B−1)
2H. Is Bsingular or nonsingular? Explain!
2I. Do the columns of Bform a linearly independent set? Explain!
