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Linear Algebra With Applications
Found in: Page 246
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Consider an m×n matrix A with Ker (A)={0}. Show that there exists an n×m matrix B such that BA=ln.

The matrix is ATA-1-AT.

See the step by step solution

Step by Step Solution

Step 1: Determine the matrix R.

Consider a m×n matrix A.

If Ker (A)={0} for a role="math" localid="1659500428667" m×n matrix A then the matrix A is invertible.

If the matrix A is invertible then the matrix role="math" localid="1659500442729" AT is invertible.

If the matrices A and B is invertible then the matrix AB is invertible.

As the value of Ker (A)={0}, by the definition the matrix A is invertible.

By the definition, the matrices AT and ATA are invertible.

Assume the matrix BA=AT A-1, simplify the matrix BA as follows.

BA=A-1lnA =A-1AT-1 ATA =A-1AT-1 ATABA=A-1lnA

Further, simplify the equation as follows.

BA=A-1lnA =A-1ABA=ln

Hence, for the matrix B=ATA-1-AT the equation BA=ln.

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