Linear Algebra

Section I. Definition 3053.27 A matrix A is skew-symmetric if A trans = −A, as in this matrix.( )0 3A =−3 0Show that n×n skew-symmetric matrices with nonzero determinants exist only foreven n.̌ 3.28 What is the smallest number of zeros, and the placement of those zeros, neededto ensure that a 4×4 matrix has a determinant of zero?̌ 3.29 If we have n data points (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x n , y n ) and want to find apolynomial p(x) = a n−1x n−1 + a n−2x n−2 + · · · + a 1x + a 0 passing through thosepoints then we can plug in the points to get an n equation/n unknown linearsystem. The matrix of coefficients for that system is called the Vandermondematrix. Prove that the determinant of the transpose of that matrix of coefficients1 1 . . . 1x 1 x 2 . . . x n222x 1 x 2 . . . x n .∣ n−1 n−1n−1x 1 x 2 . . . x n∣equals the product, over all indices i, j ∈ {1, . . . , n} with i < j, of terms of theform x j − x i . (This shows that the determinant is zero, and the linear system hasno solution, if and only if the x i ’s in the data are not distinct.)3.30 A matrix can be divided into blocks, as here,( 1 2) 03 4 00 0 −2which shows four blocks, the square 2×2 and 1×1 ones in the upper left and lowerright, and the zero blocks in the upper right and lower left. Show that if a matrixcan be partitioned as( )J Z2T =Kwhere J and K are square, and Z 1 and Z 2 are all zeroes, then |T | = |J| · |K|.̌ 3.31 Prove that for any n×n matrix T there are at most n distinct reals r suchthat the matrix T − rI has determinant zero (we shall use this result in ChapterFive).? 3.32 The nine positive digits can be arranged into 3×3 arrays in 9! ways. Find thesum of the determinants of these arrays. [Math. Mag., Jan. 1963, Q307]3.33 Show that[Math. Mag., Jan. 1963, Q237]Z 1x − 2 x − 3 x − 4x + 1 x − 1 x − 3∣x − 4 x − 7 x − 10∣ = 0.? 3.34 Let S be the sum of the integer elements of a magic square of order three andlet D be the value of the square considered as a determinant. Show that D/S isan integer. [Am. Math. Mon., Jan. 1949]