3.41 An m×n matrix has full row rank if its row rank is m, and it has full column rank if its column rank is n. (a) Show that a matrix can have both full row rank and full column rank only if it is square. (b) Prove that the linear system with matrix of coefficients A has a solution for dn's on the right side if and only if A has full row rank. .... (c) Prove that a homogeneous system has a unique solution if and only if its matrix of coefficients A has full column rank. (d) Prove that the statement "if a system with matrix of coefficients A has any solution then it has a unique solution" holds if and only if A has full column rank.
3.41 An m×n matrix has full row rank if its row rank is m, and it has full column rank if its column rank is n. (a) Show that a matrix can have both full row rank and full column rank only if it is square. (b) Prove that the linear system with matrix of coefficients A has a solution for dn's on the right side if and only if A has full row rank. .... (c) Prove that a homogeneous system has a unique solution if and only if its matrix of coefficients A has full column rank. (d) Prove that the statement "if a system with matrix of coefficients A has any solution then it has a unique solution" holds if and only if A has full column rank.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please do part A,B,C,D and please show step by step and explain
Transcribed Image Text:3.41 An mxn matrix has full row rank if its row rank is m, and it has full column
rank if its column rank is n.
(a) Show that a matrix can have both full row rank and full column rank only if
it is square.
(b) Prove that the linear system with matrix of coefficients A has a solution for
any d₁,..., dn 's on the right side if and only if A has full row rank.
(c) Prove that a homogeneous system has a unique solution if and only if its
matrix of coefficients A has full column rank.
(d) Prove that the statement “if a system with matrix of coefficients A has any
solution then it has a unique solution" holds if and only if A has full column
rank.
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