Short Answer

If the matrix $[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$ is invertible, then the matrix $[\begin{matrix} a & b \\ d & e \end{matrix}]$ must be invertible as well.

The statement is false.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Objective

The given statement is “if the matrix $[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$ is invertible, then the matrix $[\begin{matrix} a & b \\ d & e \end{matrix}]$ is invertible.” The objective is to determine whether or not the given statement is true.

Step 2: Result

Consider the matrix $A = [\begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 4 \\ 6 & 7 & 8 \end{matrix}]$ . det(A) = 5 Therefore, A is invertible.

Let $C = [\begin{matrix} 1 & 2 \\ 1 & 2 \end{matrix}]$

det(C)=0

Therefore, C is not invertible.

The matrix $[\begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 4 \\ 6 & 7 & 8 \end{matrix}]$ is invertible but the matrix $[\begin{matrix} 1 & 2 \\ 1 & 2 \end{matrix}]$ is not invertible. Therefore, the given statement is false.

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Linear Algebra With Applications

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Short Answer

If the matrix $[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$ is invertible, then the matrix $[\begin{matrix} a & b \\ d & e \end{matrix}]$ must be invertible as well.

Step by Step Solution

Step 1: Objective

Step 2: Result

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If the matrix [abcdefghi] is invertible, then the matrix [abde] must be invertible as well.

Step by Step Solution

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If the matrix $[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$ is invertible, then the matrix $[\begin{matrix} a & b \\ d & e \end{matrix}]$ must be invertible as well.