Short Answer

We are told that a certain $5 \times 5$ matrix $A$ can be written as

$A = B C$ ,

where $B$ is $5 \times 4$ and $C$ is $4 \times 5$ . Explain how you know that $A$ is not invertible.

Thus, it is proved matrix $A$ is not invertible.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given in the question.

Let matrix $A$ is $5 \times 5$ which is $A = B C$ where $B$ is $5 \times 4$ and $C$ is $4 \times 5$ .

Let assume that $c o l s p a c e (A) \subseteq c o l s p a c e (B)$ . There are different ways and one way is as the linear transformation $T_{A}$ is the composition of $T_{B}$ and $T_{C}$ gives $i m (T_{A}) \subseteq i m (T_{B})$ .

Step 2: Explanation of A is not invertible.

Take dimensions where $r a n k (A) \leq r a n k (B)$ . Now by the rank-nullity theorem applied to $B$ , gives $r a n k (B) \leq 4$ .

Which implies that $r a n k (A)$ is never 5, that is needed for $A$ to be invertible.

Hence, matrix $A$ is not invertible.

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Linear Algebra With Applications

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Short Answer

We are told that a certain $5 \times 5$ matrix $A$ can be written as

$A = B C$ ,

where $B$ is $5 \times 4$ and $C$ is $4 \times 5$ . Explain how you know that $A$ is not invertible.

Step by Step Solution

Step 1: Given in the question.

Step 2: Explanation of A is not invertible.

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Linear Algebra With Applications

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Short Answer

We are told that a certain5×5 matrixA can be written as A=BC, where Bis 5×4andC is 4×5. Explain how you know thatA is not invertible.

Step by Step Solution

Step 1: Given in the question.

Step 2: Explanation of A is not invertible.

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We are told that a certain $5 \times 5$ matrix $A$ can be written as

$A = B C$ ,

where $B$ is $5 \times 4$ and $C$ is $4 \times 5$ . Explain how you know that $A$ is not invertible.