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Question Appendix.9: Using a Counterexample in Linear Algebra Use a counterexampl...

Using a Counterexample in Linear Algebra

Use a counterexample to show that the statement is false.

If A and B are square singular matrices of order n, then A + B is a singular matrix of order n.

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Let A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} and B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} . Both A and B are singular of order 2, but

A + B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

is the identity matrix of order 2, which is not singular.

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