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

The statement is $p \Rightarrow q$ with $p:$ "$A$ invertible", $q:$ "$A$ non-singular". Prove $\sim q \Rightarrow \sim p$: if $A$ is singular then $|A| = 0$, so

$A^{-1} = \frac{\operatorname{adj} A}{|A|}$ …

Write the statement in symbolic form as $p \Rightarrow q$, where

We prove the equivalent contrapositive $\sim q \Rightarrow \sim p$: if $A$ is not non-singular, then $A$ is not invertible.

Step 1 — Translate the hypothesis $\sim q$.

If $A$ is not non-singular, then $A$ is singular, which means

Step 2 — Test whether an inverse can exist.

The inverse of a square matrix is given by

With $|A| = 0$, this expression divides by $0$, so it is undefined — $A^{-1}$ does not exist.

Step 3 — Conclude $\sim p$. …

Method: Proving an Implication via the Contrapositive (Matrix Case)

Use this whenever you're asked to "show that if [property A] then [property B]" about a matrix and the forward direction is awkward to argue directly, but the reverse (negated) direction follows immediately from a known formula.

Steps

Step 1: Write the statement as $p \Rightarrow q$

Identify $p$ = the hypothesis (e.g. "$A$ is invertible") and $q$ = the conclusion (e.g. "$A$ is non-singular, i.e. $|A|\ne0$").

Step 2: Form and state the contrapositive $\sim q \Rightarrow \sim p$

Negate and swap: "if $A$ is singular ($|A|=0$), then $A$ is not invertible." Confirm this is logically equivalent to the original — proving one proves the other.

Step 3: Assume $\sim q$ and test the defining formula …

Common Mistakes

Mistake 1: Arguing directly and circularly instead of via the contrapositive

Reasoning "since $A$ is invertible, $A^{-1}=\dfrac{\operatorname{adj}A}{|A|}$ exists, so $|A|\ne 0$" skips straight to a forward algebraic argument without ever stating $p\Rightarrow q$ and its contrapositive. Why it's wrong: the exercise specifically names the contrapositive technique — jumping to a bare forward statement misses the reasoning skill being tested and can leave the logical structure incompletely justified. Correct approach: explicitly name $p$ and $q$, state the contrapositive $\sim q\Rightarrow\sim p$, and prove that.

Mistake 2: Confusing "singular" and "non-singular" …