159 3.1 SOLUTIONS Notes: f time is needed for other topics, this chapter may be omitted. Section 5.2 contains enough information about determinants to support the discussion there of the characteristic polynomial of a matrix. In section 5.1, some exercises in this section provide practice in computing determinants, while others allow the student to discover the properties of determinants which will be studied in the next section. Determinants are developed through the cofactor expansion, which is given in Theorem 1. Exercises 33–36 in this section provide the first step in the inductive proof of Theorem 3 in the next section. A “Checkpoint” in the Study Guide leads students to discover that if the kth column of the identity matrix is replaced by a vector x, then the determinant of the resulting matrix is the kth entry of x. This idea is used in the proof of Cramer’s Rule, in Section 3.3. 1. Expand across along the first row: 3043222232323043( 13)4(10)1510105051=−+=−+=−−− Expand down the second column: 1 22 23 2304223434232( 1)0( 1)3( 1)53( 3)5( 2)1010122051+++= −⋅+ −⋅+ −⋅=−−−=−−− 2. Expand across the first row: 0513040434300515(4)1(22)2412124241−−−=−+= −+= Expand down the second column: 1 22 23 2051400101430( 1)5( 1)( 3)( 1)45(4)3( 2)4( 4)2212140241+++−= −⋅+ −⋅ −+ −⋅= −−−−−= 3. Expand across the first row: 2431232313122( 4)32( 9)4( 5)(3)(11)5411114141−=− −+=−+−+= −−−− 160 CHAPTER 3 • Determinants Expand down the second column: 1 22 23 2243322323312( 1)( 4)( 1)1( 1)44( 5)1( 5)4( 5)5111132141+++−= −⋅ −+ −⋅+ −⋅=−+−−−= −−−− 4 . Expand across the first row: 1351121212111351( 2)3(1)5(5)20423234342=−+=−−+= Expand down the second column: 1 22 23 213...
