Show that if a matrix A is invertible, then A is non-singular. - Vidyadip

Question

Show that if a matrix A is invertible, then A is non-singular.

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Answer

Writing the above statement in symbolic form, we have p ⇒ q, where, p is “matrix A is invertible” and q is “A is non singular”
Instead of proving the given statement, we prove its contrapositive statement, i.e., if A is not a non singular matrix, then the matrix A is not invertible.
If A is not a non singular matrix, then it means the matrix A is singular, i.e.,
|A| = 0
Then $\mathrm{A}^{-1}=\frac{a d j \mathrm{A}}{| \mathrm{Al}} \text { does not exist as } \mathrm{|A|}=0$
Hence, A is not invertible.
Thus, we have proved that if A is not a non singular matrix, then A is not invertible.
i.e -q = -p
Hence, if a matrix A is invertible, then A is non singular.

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