Chapter 14. Three-by-Three Matrices and Determinants. A 3 3 matrix looks like a 11 a 12 a 13 A = a 21 a 22 a 23

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1 1 Chapter 14. Three-by-Three Matrices and Determinants A 3 3 matrix looks like a 11 a 12 a 13 A = a 21 a 22 a 23 = [a ij ] a 31 a 32 a 33 The nmber a ij is the entry in ro i and colmn j of A. Note that A has three ro vectors: ro i (A) = (a i1, a i2, a i3 ), i = 1, 2, 3 as ell as three colmn vectors: a 1j col j (A) = a 2j, j = 1, 2, 3. a 3j The prodct of to matrices A = [a ij ] and B = [b ij ] is the matrix AB = [c ij ] hose entry in ro i and colmn j is the dot prodct The identity matrix is c ij = ro i (A), col j (B) = a i1 b 1j + a i2 b 2j + a i3 b 3j I = It has the property that AI = IA = A for every 3 3 matrix A. The inverse of a matrix A is the matrix A 1 sch that AA 1 = A 1 A = I. As ith the 2 2 case, not all matrices are invertible. Later, e ill see exactly hen A 1 exists, and e ill have a formla for A 1 hen it does exist. Finally, the transpose of A is the matrix A T sch that ro i (A T ) = col i (A). That is, to get A T yo flip A abot the diagonal. A 3 3 matrix A moves vectors arond in space. More precisely, A can be vieed as the linear transformation from R 3 to R 3 hich sends the vector = (x, y, z) to the vector ro 1 (A), a 11 x + a 12 y + a 13 z A = ro 2 (A), = a 21 x + a 22 y + a 23 z. ro 3 (A), a 31 x + a 32 y + a 33 z

2 2 Recall that the standard basis vectors in R 3 are e 1 = (1, 0, 0), e 2 = (0, 1, 0), e 3 = (0, 0, 1). The Most Important Thing, again, is that the j th colmn of A is hat A does to e j. In other ords, col j (A) = Ae j (M.I.T.) Example: The matrix cos θ 0 sin θ A = sin θ 0 cos θ is a rotation by θ arond the y axis. Yo can see right aay that the y axis is fixed, since Ae 2 = e 2. (1) The corner entries of A give a rotation in the xz plane. Since for small θ the z- coordinate of Ae 1 is positive, the rotation is clockise by θ, as the positive y-axis points toards the vieer. Example: The matrix B = is again a rotation matrix, bt no the axis is not so obvios. First note that Be 1 = e 2, Be 2 = e 3, Be 3 = e 1 so that B permtes the vertices of the frst qadrant nit cbe (hose vertices have components either 0 or 1). Note also that that B 3 = I and that B fixes the vector = (1, 1, 1): B =, here = (1, 1, 1). (2) No yo can see that B is rotation by 2π/3 abot the diagonal of the cbe throgh the opposite vertices 0 and. In both of these examples, the axis of rotation is throgh an eigenvector ith eigenvale λ = 1, as expressed in eqations (1) and (2). To find eigenvales and eigenvectors, e need some comptational tools. The main tool is the determinant, hich yo can compte as follos.

3 If A is a 3 3 matrix, e define A ij to be the 2 2 sbmatrix of A obtained by deleting ro i and colmn j. For example, if i = 1 and j = 3, then [ ] A 13 = a 21 a 22 a21 a = 22. a a 31 a a 32 The determinant of A is defined to be det A = a 11 det A 11 a 12 det A 12 + a 13 det A 13 (3) Note the connnection ith the cross-prodct: If e rite A = v, here, v, are the ro vectors of A, then det(a) = (v ). (4) Formla (4) also holds hen A = [ v ], that is, hen, v, are the colmn vectors of A. Formla (3) is an expansion along the top ro. In fact, yo can expand along any ro or colmn. For example, yo can also compte det A as 3 det A = a 21 det A 21 + a 22 det A 22 a 23 det A 23, det A = a 13 det A 13 a 23 det A 23 a 33 det A 33, The rle for the signs is given in the folloing pictre: (second ro) (third colmn). That is, the term a ij det A ij gets sign ( 1) i+j. No matter hich ro or colmn yo choose to expand, the reslt ill be det A = a 11 a 22 a 33 a 12 a 21 a 33 a 13 a 22 a 31 a 11 a 23 a 32 + a 12 a 23 a 31 + a 13 a 21 a 32. (5)

4 4 Basic Properties of the Determinant: 1. det(ab) = (det A)(det B) 2. det(a) = det(a T ). 3. A matrix A has an inverse A 1 exactly hen det(a) 0, in hich case det(a 1 ) = 1 det(a). 4. If yo sitch to ros of A, the determinant changes by a sign. The same holds for the colmns. For example, det = det v and det [ v ] = det [ v ] v 5. If yo add a mltiple of one ro to another ro the determinant is nchanged. The same holds for the colmns. For example, + c det v = det v. 6. det(a) is the volme of the parallelopiped spanned by the ro (or colmn) vectors of A. A parallelopiped is a six-sided box ith opposite slides parallel. The parallelogram spanned by three vectors, v, is the collection of points c 1 + c 2 v + c 3, here 0 c i 1 for i = 1, 2, 3. Property 6 of the determinant comes directly from Eqation (4), hich tells s that det A = (v ) = v cos θ, here θ is the angle beteen and v. No v is the area of the parallelogram P spanned by v and, and P can be taken as the base of the parallelopiped spanned by, v,. The volme of the latter is the base times the height, and the height is cos θ, proving that det A is the asserted volme.

5 5 Formla for the inverse of a matrix: The determinant can be sed to find the inverse of a matrix. Assme that det(a) 0. Let 1, 2, 3 be the ros of A. Take cross-prodcts as follos: v 1 = 2 3, v 2 = 3 1, v 3 = 1 2, and let B be the matrix hose colmns are v 1, v 2, v 3. So e have Since 1 A = 2, B = [ v 1 v 2 ] v 3. 3 det(a) = 1, 2 3 = 3, 1 2 = 2, 3 1 and i, j k = 0 if i = j or i = k, e have AB = det(a)i. Hence e get the formla A 1 = 1 B. (6) det A Of corse, this only makes sense if det(a) 0. If det(a) = 0 then A 1 does not exist. If yo rite ot Formla (6) in terms of the original ro vectors i, it looks like Note the cyclic pattern again. 2 3 A 1 = 1 det A Example: We compte: A = = ( 22, 14, 8), 3 1 = ( 51, 33, 20), 1 2 = ( 19, 13, 8), and det(a) = 1, 2 3 = 4, so A 1 =

6 6 Exercise 14.1 Find the matrices R x, R y, R z that rotate by π abot the x, y, z axes respectively. Exercise 14.2 Find the matrices R xy, R yz, R zx that reflect abot the xy, yz, zx planes, respectively. Exercise 14.3 Find the determinants of the folloing matrices. Do not se a calclator and sho all of yor ork a 0 0 (a) A = (b) A = (c) A = 0 b c a x y x x + 1 x + 2 (d) A = 0 b z (e) A = (f) A = x + 3 x + 4 x c x + 6 x + 7 x + 8 Exercise 14.4 Make p to vectors, v in R 3, choose to scalars a and b, and let = a + bv. Compte the determinant of the matrix A hose colmns are, v,. The anser does not depend on yor choices. Exercise 14.5 Dra the parallelopiped spanned by e 1, e 1 +e 2, e 1 +e 2 +e 3. Then compte its volme. Exercise 14.6 Find the inverses of the folloing matrices. Do not se a calclator and sho all of yor ork x z (a) A = (b) A = 0 1 y (c) A = Exercise 14.7 There are six ays to permte the nmbers 1, 2, 3. For each permtation σ, let A σ be the matrix hich sends e i to e σ(i). (a) Write don the six matrices A σ and compte their determinants. (b) Each permtation σ corresponds to the term a 1σ(1) a 2σ(2) a 3σ(3) in the expanded determinant (5). What is the relation beteen the sign of this term and det(a σ )? (c) Each permtation corresponds to a symmetry of an eqilateral triangle ith vertices labelled 1, 2, 3. Three of the six symmetries are rotations of the triangle, and the other three are reflections. What is the relation beteen this dichotomy and det(a σ )?