Linear Algebra

224 Chapter Three. Maps Between Spaceš 3.27 Prove that the diagonal matrices form a subspace of M n×n. What is itsdimension?3.28 Does the identity matrix represent the identity map if the bases are unequal?3.29 Show that every multiple of the identity commutes with every square matrix.Are there other matrices that commute with all square matrices?3.30 Prove or disprove: nonsingular matrices commute.̌ 3.31 Show that the product of a permutation matrix and its transpose is an identitymatrix.3.32 Show that if the first and second rows of G are equal then so are the first andsecond rows of GH. Generalize.3.33 Describe the product of two diagonal matrices.3.34 Write ( )1 0−3 3as the product of two elementary reduction matrices.̌ 3.35 Show that if G has a row of zeros then GH (if defined) has a row of zeros.Does that work for columns?3.36 Show that the set of unit matrices forms a basis for M n×m.3.37 Find the formula for the n-th power of this matrix.( )1 11 0̌ 3.38 The trace of a square matrix is the sum of the entries on its diagonal (itssignificance appears in Chapter Five). Show that trace(GH) = trace(HG).̌ 3.39 A square matrix is upper triangular if its only nonzero entries lie above, oron, the diagonal. Show that the product of two upper triangular matrices is uppertriangular. Does this hold for lower triangular also?3.40 A square matrix is a Markov matrix if each entry is between zero and oneand the sum along each row is one. Prove that a product of Markov matrices isMarkov.̌ 3.41 Give an example of two matrices of the same rank with squares of differingrank.3.42 Combine the two generalizations of the identity matrix, the one allowing entiresto be other than ones, and the one allowing the single one in each row andcolumn to be off the diagonal. What is the action of this type of matrix?3.43 On a computer multiplications are more costly than additions, so people areinterested in reducing the number of multiplications used to compute a matrixproduct.(a) How many real number multiplications are needed in formula we gave for theproduct of a m×r matrix and a r×n matrix?(b) Matrix multiplication is associative, so all associations yield the same result.The cost in number of multiplications, however, varies. Find the associationrequiring the fewest real number multiplications to compute the matrix productof a 5×10 matrix, a 10×20 matrix, a 20×5 matrix, and a 5×1 matrix.(c) (Very hard.) Find a way to multiply two 2 × 2 matrices using only sevenmultiplications instead of the eight suggested by the naive approach.? 3.44 If A and B are square matrices of the same size such that ABAB = 0, doesit follow that BABA = 0? [Putnam, 1990, A-5]