Matrix Operations and Properties: Triangular Matrices, Matrix Rank, and Symmetry
Explore various matrix properties and operations including reflections, diagonal matrices, triangular matrices, matrix ranks, trace of a matrix, and symmetry. Learn about how matrices are affected by different transformations and how certain matrix properties relate to each other.
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Presentation Transcript
58. Let ? = ?180, and let ? be the matrix that reflects ?2about the x-axis; that is, ? =1 0 ? = 1 0 0 1 0 1 Compute ??, and describe geometrically how a vector ? is affected by multiplication by ??.
60. Let ? be an ? ? matrix. If ? and ? are ? ? diagonal matrices, then ?? is a diagonal matrix whose ?th columns is ????????.
59. A square matrix ? is called lower triangular if the ?,? -entry of ? is zero whenever ? < ?. Prove that if ? and ? are both ? ? lower triangular matrices, then ?? is also a lower triangular matrix. 61. A square matrix ? is called upper triangular if the ?,? -entry of ? is zero whenever ? > ?. Prove that if ? and ? are both ? ? upper triangular matrices, then ?? is also an upper triangular matrix.
2 1 3 3 7 1 1 1 1 3 Find a nonzero 4 2 matrix ? with rank 2 such that ?? = ? 1 2 1 5 62. Let ? = . 4 4
63. Find an example of ? ? matrices ? and ? such that ?? = ?, but ?? ?. 64. Let ? and ? be ? ? matrices. Prove and disprove that the ranks of ?? and ?? are equal.
65. Recall the definition of trace of a matrix. Prove that if ? is an ? ? matrix and ? is an ? ? matrix, then ????? AB = ????? BA .
66. Let 1 ?,? ? be integers, and let ? be the ? ? matrix with 1 as the ?,? -entry and 0s elsewhere. Let ? be any ? ? matrix. Describe ?? in terms of the entries of ?.
68. (a) Let ? and ? be symmetric matrices of the same size. Prove that ?? is symmetric if and only if ?? = ??. (b) Find symmetric 2 2 matrices ? and ? such that ?? ??.
