How does one show that the following matrix factorization inequality holds in $M_{n} (\mathcal{A})^{+}$,
$$(a_{i}^{*}a^{*}aa_{j}) \leq ||a^{*}a|| \cdot (a_{i}^{*}a_{j})$$
Notation. Let $M_{n} (\mathcal{A})$ denote the space of $n \times n$ matrices with entries from $\mathcal{A}$, where $\mathcal{A}$ is an unital C*-algebra. Let $M_{n} (\mathcal{A})^{+} \subset M_{n} (\mathcal{A})$ denote the positive elements in $M_{n} (\mathcal{A})$.
I am using Vern Paulsen's text Completely Bounded Maps and Operator Algebras, and have been stuck on proving the aforementioned inequality for an embarrassing amount of time.
For every positive element $a$ one has $a\leq\Vert a\Vert I$