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Linear Algebra With Applications
Found in: Page 144
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

We are told that a certain5×5 matrixA can be written as


where Bis 5×4andC is 4×5. Explain how you know thatA is not invertible.

Thus, it is proved matrixA is not invertible.

See the step by step solution

Step by Step Solution

Step 1: Given in the question.

Let matrix Ais 5×5which is A=BCwhereB is 5×4andC is4×5 .

Let assume thatcolspace(A)colspace(B) . There are different ways and one way is as the linear transformationTA is the composition of TBandTC gives im(TA)im(TB).

Step 2: Explanation of A is not invertible.

Take dimensions whererank(A)rank(B) . Now by the rank-nullity theorem applied to B, givesrank(B)4.

Which implies thatrank(A) is never 5, that is needed for Ato be invertible.

Hence, matrixAis not invertible.

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