24. We can writef(x) =x3-x=x(x2-1) =x(x-1)(x+ 1).The equationf(x) = 0 has three solutions,x= 0,1,-1, so thatf(x) fails to be invertible.25. Invertible, with inversex1x2=3√y1y226. Invertible, with inversex1x2=3√y2-y1y188
ISM:Linear AlgebraSection 2.3Figure 2.40: for Problem 188.8.131.52. This transformation fails to be invertible, since the equationx1+x2x1x2=01has nosolution.28. We are asked to find the inverse of the matrixA=221383-16-3-2-289725431.We find thatA-1=1-29-25-25-22604-941-112-91780222.T-1is the transformation fromR4toR4with matrixA-1.29. Use Fact 2.3.3:11112k14k2-I-I→11101k-103k2-1-II-3(II)→102-k01k-100k2-3k+ 2The matrix is invertible if (and only if)k2-3k+ 2 = (k-2)(k-1) = 0, in which casewe can further reduce it toI3. Therefore, the matrix is invertible ifk= 1 andk= 2.30. Use Fact 2.3.3:01b-10c-b-c0----→I↔II-10c01b-b-c0----→÷(-1)10-c01b-b-c0+b(I) +c(II)-→10-c01b000This matrix fails to be invertible, regarless of the values ofbandc.31. Use Fact 2.3.3; first assume thata= 0.89
Chapter 2ISM:Linear Algebra0ab-a0c-b-c0swap :I↔II→-a0c0ab-b-c0÷(-a)→10-ca0ab-b-c0+b(I)→10-ca0ab0-c-bca÷a→10-ca01ba0-c-bca+c(II)→10-ca01ba000Now consider the case whena= 0:00b00c-b-c0swap :I↔III→-b-c000c00b: The second entry on the diagonal of rrefwill be 0.It follows that the matrix0ab-a0c-b-c0fails to be invertible, regardless of the valuesofa, b, andc.32. Use Fact 2.3.6.IfA=abcdis a matrix such thatad-bc= 1 andA-1=A, thenA-1=1ad-bcd-b-ca=d-b-ca=abcd, so thatb= 0, c= 0, anda=d.The conditionad-bc=a2= 1 now implies thata=d= 1 ora=d=-1.This leaves only two matricesA, namely,I2and-I2. Check that these two matrices doindeed satisfy the given requirements.33. Use Fact 2.3.6.The requirementA-1=Ameans that-1a2+b2-a-b-ba=abb-a. This is the caseif (and only if)a2+b2= 1.34. a. By Fact 2.3.3,Ais invertible if (and only if)a, b, andcare all nonzero. In this case,A-1=1a0001b0001c.90
ISM:Linear AlgebraSection 2.3b. In general, a diagonal matrix is invertible if (and only if) all of its diagonal entries arenonzero.35. a.Ais invertible if (and only if) all its diagonal entries,a, d, andf, are nonzero.b. As in part (a): if all the diagonal entries are nonzero.c. Yes,A-1will be upper triangular as well; as you construct rref[A...In], you will performonly the following row operations:•divide rows by scalars•subtract a multiple of thejth row from theith row, wherej > i.