Animated Solution for Mathematics - Matrices and Determinants: If $A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$ and $A\text{ adj }A=A{A}^T$, then $5a+b$ is equal to:

Welcome, future engineer! Today, we are going to unravel a beautiful problem that tests not just your ability to calculate, but your ability to see the hidden structure within matrices.

We are given a matrix $A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$ and a fascinating equation: $A\text{ adj }A=A{A}^T$. Our mission is to find the value of $5a+b$.

At first glance, this might look like a standard matrix problem, but there is a trap here: if you try to calculate the adjoint of $A$ directly, you are walking into a long, tedious path. Let us take a smarter route.

In the world of JEE mathematics, there is a golden rule for adjoint matrices: $A\text{ adj }A=|A|I$. This property is a lifesaver.

It tells us that $A$ multiplied by its adjoint is simply the determinant of $A$ multiplied by the identity matrix. For our two-by-two matrix, the determinant is:

Now, our left-hand side becomes a simple diagonal matrix:

Now, let us look at the right-hand side: $A{A}^T$. We know $A^T=\left[\begin{array}{cc}5a& 3\\ -b& 2\end{array}\right]$.

When we perform the multiplication $A{A}^T$, we get:

Now, we equate the two matrices. For them to be equal, every corresponding element must match.

Looking at the off-diagonal elements, we get $15a-2b=0$, which means:

Looking at the bottom-right element, we get $10a+3b=13$. Substituting $b$ into this equation, we get:

This simplifies to $65a=26$, or $a=\frac{2}{5}$. With $a=\frac{2}{5}$, we find $b=\frac{15}{2}\cdot \frac{2}{5}=3$.

Finally, the expression $5a+b$ becomes $5(\frac{2}{5})+3=2+3=5$.

And there it is—a perfect, clean result. The final answer is 5.