Solutions to ExercisesCollege of the RedwoodsMathematics DepartmentMath Math 45 — Linear AlgebraQuiz #8—Linear AlgebraDavid ArnoldCopyrightc 2000 [email protected] Revision Date: November 13, 2001 Version 1.00Directions: Place your solution in the space provided. No calculators allowed!Exercise 1. Use elimination to compute the following determinant. Show all of your work, connectingconsecutive computations with equal signs.01 21 −1222−3=Exercise 2. Suppose that matrix A is a 4 × 4 matrix. Answer each of the following questions, assuming|A| =3.(a) |AT| =(b) |A−1| =(c) |3A| =Exercise 3. Each of the following determinants is easily calculated using one or more of the 10 propertiesof the determinant proposed by Strang. In each exercise, compute the determinant without resorting toelimination. Give a handwritten reason for your conclusion; i.e., state which properties you use to arrive atyour answer.(a)00 302−114 5=(b)aca+ cbdb+ dcec+ e=(c)010001100=Solutions to ExercisesExercise 1. Swap rows 1 and 201 21 −1222−3=1 −1201 222−3Subtract 2 times row 1 from row 3.= −1 −1201 204−7Subtract 4 times row 2 from row 3.=1 −1201 200−15= −(1)(1)(15)=15Exercise 1Exercise 2(a) The determinant of the transpose equals the determinant of the original matrix. Therefore,|AT| = |A| =3.Exercise 2(b) Because AA−1= I, we can write|AA−1| = |I | =1.The determinant of a product is the product of the determinants. Thus,|A||A−1| =1|A−1| =1|A|=13.Exercise 2(c) Every time we multiply a row by 3, we multiply the value of the determinant by 3. But 3Amultiplies each of 4 rows of A (A is 4 × 4) by 3. Thus,|3A| =34|A| =81· 3 = 243Exercise 3(a) Swapping rows 1 and 3 negates the determinant00 302−114 5=14 502−100 3The determinant of a triangular matrix is the product of its diagonal elements.= (1)(2)(3)=6.Exercise 3(b) A determinant is linear in its columns. Thus,aca+ cbdb+ dcec+ e=acabdbcec+accbddceeIf a matrix has two equal columns, its determinant is zero.=0+0=0Exercise 3(c) Swapping rows 1 and 3 negates the determinant010001100= −100001010Swapping row 2 and 3 negates
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