Linear Algebra/Exploration/Solutions

Solutions Problem 1 Evaluate the determinant of each. <li> <li> <li> Answer <li> <li> <li> Problem 2 Evaluate the determinant of each. <li> <li> <li> Answer <li> <li> <li> Problem 3 Verify that the determinant of an upper-triangular matrix is the product down the diagonal. Do lower-triangular matrices work the same way? Answer For the first, apply the formula in this section, note that any term with a , , or is zero, and simplify. Lower-triangular matrices work the same way. Problem 4 Use the determinant to decide if each is singular or nonsingular. <li> <li> <li> Answer <li> Nonsingular, the determinant is . <li> Nonsingular, the determinant is . <li> Singular, the determinant is . Problem 5 Singular or nonsingular? Use the determinant to decide. <li> <li> <li> Answer <li> Nonsingular, the determinant is . <li> Singular, the determinant is . <li> Singular, the determinant is . Problem 6 Each pair of matrices differ by one row operation. Use this operation to compare with . <li> <li> <li> Answer <li> via <li> via <li> via Problem 7 Show this. Answer Using the formula for the determinant of a matrix we expand the left side and by distributing we expand the right side. VandermondeNow we can just check that the two are equal. (Remark. This is the case of Vandermonde's determinant which arises in applications). Problem 8 Which real numbers make this matrix singular? Answer This equation has roots and . Problem 9exer:ThreeByThreeDetForm Do the Gaussian reduction to check the formula for matrices stated in the preamble to this section. is nonsingular iff Answer We first reduce the matrix to echelon form. To begin, assume that and that . This step finishes the calculation. Now assuming that and , the original matrix is nonsingular if and only if the entry above is nonzero. That is, under the assumptions, the original matrix is nonsingular if and only if , as required. We finish by running down what happens if the assumptions that were taken for convienence in the prior paragraph do not hold. First, if but then we can swap and conclude that the matrix is nonsingular if and only if either or . The condition " or " is equivalent to the condition "". Multiplying out and using the case assumption that to substitute for gives this. Since , we have that the matrix is nonsingular if and only if . Therefore, in this and case, the matrix is nonsingular when . The remaining cases are routine. Do the but case and the and but case by first swapping rows and then going on as above. The , , and case is easy— that matrix is singular since the columns form a linearly dependent set, and the determinant comes out to be zero. Problem 10 Show that the equation of a line in thru and is expressed by this determinant. Answer Figuring the determinant and doing some algebra gives this. Note that this is the equation of a line (in particular, in contains the familiar expression for the slope), and note that and satisfy it. Problem 11 Many people know this mnemonic for the determinant of a matrix: first repeat the first two columns and then sum the products on the forward diagonals and subtract the products on the backward diagonals. That is, first write and then calculate this. <li> Check that this agrees with the formula given in the preamble to this section. <li> Does it extend to other-sized determinants? Answer <li> The comparison with the formula given in the preamble to this section is easy. <li> While it holds for matrices it does not hold for matrices. An example is that this matrix is singular because the second and third rows are equal but following the scheme of the mnemonic does not give zero. Problem 12cross product The cross product of the vectors is the vector computed as this determinant. Note that the first row is composed of vectors, the vectors from the standard basis for . Show that the cross product of two vectors is perpendicular to each vector. Answer The determinant is . To check perpendicularity, we check that the dot product with the first vector is zero and the dot product with the second vector is also zero. Problem 13 Prove that each statement holds for matrices. <li> The determinant of a product is the product of the determinants . <li> If is invertible then the determinant of the inverse is the inverse of the determinant . similarMatrices and are similar if there is a nonsingular matrix such that . (This definition is in Chapter Five.) Show that similar matrices have the same determinant. Answer <li> Plug and chug: the determinant of the product is this while the product of the determinants is this. Verification that they are equal is easy. <li> Use the prior item. That similar matrices have the same determinant is immediate from the above two: . Problem 14 Prove that the area of this region in the plane x200px is equal to the value of this determinant. Compare with this. Answer One way is to count these areas x200px by taking the area of the entire rectangle and subtracting the area of the upper-left rectangle, the upper-middle triangle, the upper-right triangle, the lower-left triangle, the lower-middle triangle, and the lower-right rectangle . Simplification gives the determinant formula. This determinant is the negative of the one above; the formula distinguishes whether the second column is counterclockwise from the first. Problem 15 Prove that for matrices, the determinant of a matrix equals the determinant of its transpose. Does that also hold for matrices? Answer The computation for matrices, using the formula quoted in the preamble, is easy. It does also hold for matrices; the computation is routine. Problem 16 Is the determinant function linear — is ? Answer No. Recall that constants come out one row at a time. This contradicts linearity (here we didn't need , i.e., we can take to be the zero matrix). Problem 17 Show that if is then for any scalar . Answer Bring out the 's one row at a time. Problem 18 Which real numbers make singular? Explain geometrically. Answer There are no real numbers that make the matrix singular because the determinant of the matrix is never , it equals for all . Geometrically, with respect to the standard basis, this matrix represents a rotation of the plane through an angle of . Each such map is one-to-one — for one thing, it is invertible. ? Problem 19 If a third order determinant has elements , , ..., , what is the maximum value it may have? Haggett Saunders 1955 Answer This is how the answer was given in the cited source. Let be the sum of the three positive terms of the determinant and the sum of the three negative terms. The maximum value of is The minimum value of consistent with is Any change in would result in lowering that sum by more than . Therefore the maximum value for the determinant and one form for the determinant is References Vern (proposer) Haggett F. W. (solver) Saunders Elementary problem 1135 American Mathematical Monthly American Mathematical Society 62 5 Apr. 1955 257