Linear Algebra: Multiplication by Special Matrices

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SUMMARY

The discussion focuses on the multiplication of matrices by special matrices, specifically the matrices S and T derived from the identity matrix I. The matrix S is defined as S = I + cEij, where Eij is an n × n matrix with a 1 at the (i, j) position, and T modifies I by zeroing out the (i, i) and (j, j) entries while swapping the (i, j) and (j, i) entries. The user is tasked with computing the products SA, AS, TA, and AT for an n × n matrix A to understand the effects of these operations on the rows and columns of A.

PREREQUISITES
  • Understanding of matrix operations, specifically multiplication.
  • Familiarity with identity matrices and their properties.
  • Knowledge of special matrices, such as Eij.
  • Basic concepts of linear algebra, including scalar multiplication.
NEXT STEPS
  • Research the properties of special matrices in linear algebra.
  • Learn about the effects of matrix multiplication on row and column transformations.
  • Explore the implications of modifying identity matrices in matrix operations.
  • Study examples of matrix multiplication involving Eij and similar constructs.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking to enhance their understanding of matrix operations and transformations.

robbie11
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Im working on this problem and got stuck i would appreciate any help...

Let Eij be the n × n matrix with 1 at the (i, j)th place and zero elsewhere.
For a scalar c in K, put S = I + cEij . And let T be the n × n
matrix obtained from the identity matrix as follows. In I replace 1 at
the (i, i)th and (j, j)th entries by zero. Replace zeros at the (i, j)th and
(j, i)th entry by 1. Suppose A is an n × n matrix. Compute SA, AS,
TA, and AT to conclude what does the multiplication on the left or
right by S or T do to the row or columns of A.

robbie
 
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Welcome to PF!

Hi robbie! Welcome to PF!: :smile:

(try using the X2 tag just above the Reply box :wink:)

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:

(if you want to write the matrices out, either use the CODE tag, 3 to the left of the X2 tag, or use LaTeX and http://www.physics.udel.edu/~dubois/lshort2e/node54.html#SECTION00830000000000000000" )
 
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