Let's solve this step by step using inverse matrices.

Problem Breakdown

• There are 48 coins (nickels, dimes, and quarters):
  $n + d + q = 48$
  where $n$, $d$, and $q$ are the numbers of nickels, dimes, and quarters.

• The total value of the coins is $5.40:
  $0.05n + 0.10d + 0.25q = 5.40$.

• The value of the dimes is $0.55 less than the value of the nickels:
  $0.10d = 0.05n - 0.55$.

Matrix Representation

We aim to represent this system of equations as $A \mathbf{x} = \mathbf{b}$, solve for $\mathbf{x}$ using the inverse of $A$.

1. Rewrite the equations:

   • $n + d + q = 48$
   • $0.05n + 0.10d + 0.25q = 5.40$
   • $-0.05n + 0.10d = -0.55$.

2. In matrix form:

   1 & 1 & 1 \\ 0.05 & 0.10 & 0.25 \\ -0.05 & 0.10 & 0 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} n \\ d \\ q \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 48 \\ 5.40 \\ -0.55 \end{bmatrix}.$$

Solving Using Inverse Matrices

We solve using $\mathbf{x} = A^{-1} \mathbf{b}$.

Step 1: Find $A^{-1}$

The inverse of a $3 \times 3$ matrix can be computed using standard techniques or a calculator.

Step 2: Multiply $A^{-1}$ by $\mathbf{b}$

This gives the solution vector $\mathbf{x}$, where: