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Q33E

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Found in: Page 108

### Linear Algebra With Applications

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

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

The statement is false.

See the step by step solution

## Step 1: Objective

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

## Step 2: Result

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

Let $C=\left[\begin{array}{cc}1& 2\\ 1& 2\end{array}\right]$

det(C)=0

Therefore, C is not invertible.

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