89%(28)25 out of 28 people found this document helpful
This preview shows page 1 - 3 out of 9 pages.
1.(1 point) Suppose that:A=1-4-44andB=-25-34-11225Given the following descriptions, determine the following ele-mentary matrices and their inverses.a. The elementary matrixE1multiplies the first row of A by1/2.E1=,E-11=b. The elementary matrixE2multiplies the second row of Aby -2.E2=,E-12=c. The elementary matrixE3switches the first and secondrows of A.E3=,E-13=d. The elementary matrixE4adds 5 times the first row of Ato the second row of A.E4=,E-14=e. The elementary matrixE5multiplies the second row of Bby 1/5.E5=,E-15=f. The elementary matrixE6multiplies the third row of B by-2.E6=,E-16=g. The elementary matrixE7switches the first and third rowsof B.E7=,E-17=h. The elementary matrixE8adds 3 times the third row of Bto the second row of B.E8=,E-18=Solution:a.E1is obtained by multiplying by12the first rowof the identity matrixI=1001. This givesE1=12001The inverse ofE1is obtained by multiplying the first row of theidentity by 2. This givesE-11=2001.b.E2is obtained by multiplying by-2 the second row ofthe identity matrixI=1001. This givesE2=100-2The inverse ofE2is obtained by dividing the first row of theidentity by-2. This givesE-12=1001-2.c.E3is obtained by switching the first and second rows ofthe identity matrixI=1001. This givesE3=0110The inverse ofE3is obtained by switching the first and secondrows of the identity . This givesE-13=0110.Note thatE3=E-13. This is always the case for permutationmatrices, that is, matrices that are obtained by permuting therows of the identity matrix.d.E4is obtained by performing the row operationr2←r2+5r1on the identity matrixI=1001. This givesE4=1051The inverse ofE4is obtained by performing the row op-erationr2←r2-5r1on the identity matrix.This givesE-14=10-51.e.E5is obtained by multiplying by15the second row of theidentity matrixI=100010001. This givesE5=1000150001The inverse ofE5is obtained by multiplying the second row ofthe identity by 5. This givesE-15=100050001.f.E6is obtained by multiplying by-2 the third row of theidentity matrixI=100010001. This givesE6=10001000-2The inverse ofE6is obtained by dividing the third row of theidentity by-2. This givesE-16=100010001-2.g.E7is obtained by switching the fist and the third row of the1
identity matrixI=100010001. This givesE7=001010100The inverse ofE7is obtained by switching the first and thirdrow of the identity . This givesE-17=001010100.