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Understanding the determinant of a matrix is crucial when examining whether a matrix can be inverted. The determinant can be thought of as a scalar value that provides essential information about the matrix, including the volume distortion during the linear transformation it represents and whether the matrix has an inverse.

For a 3x3 matrix like in our exercise, the determinant evaluates to a single number obtained by a specific arithmetic involving its elements. Here's a simplified formula for a 3x3 matrix: \[\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\], where \(a_{ij}\) are the elements of the matrix. If the determinant is zero, the matrix cannot be inverted, which is why it's the first property we check in our exercise. In the given problem, since we know \(a\) is non-zero, we concluded that the determinant is non-zero, hence the matrix is invertible.

A diagonal matrix is a type of matrix where the entries outside the main diagonal are all zero. The main diagonal is from the top left to the bottom right of the matrix. In our exercise, the matrix provided is indeed a diagonal matrix because all entries except those on the main diagonal are zero.

This kind of matrix is significant because it simplifies many operations. For example, computing the power of a diagonal matrix or finding its inverse is straightforward. The inverse of a diagonal matrix, if it exists (which is when all the diagonal elements are non-zero), is simply another diagonal matrix where each diagonal element is the reciprocal of the original matrix's corresponding diagonal element.

To illustrate using our exercise, the inverse of the matrix is found by taking the reciprocal of each non-zero diagonal element which gives us: \[\left[\begin{array}{ccc}1/a & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]\], thus affirming the matrix's diagonal nature.

The identity matrix plays a fundamental role in linear algebra. This matrix serves as the multiplicative identity in the matrix world, similar to how the number 1 does in ordinary arithmetic. It is denoted by \(I\) and is a diagonal matrix with ones on the main diagonal and zeros everywhere else.

When you multiply any matrix by the identity matrix, the result is the original matrix. The same goes for matrix inversion; when the inverse of a matrix is multiplied by the original matrix, the result should be the identity matrix. This is why finding the inverse is synonymous with solving a matrix equation: \(A^{-1}A = I\).

In the provided exercise, once we compute the inverse of the matrix for the special case where \(a=1\), we obtain the identity matrix, showing that the inverse operation preserves the identity: \[\left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right] = I\]. This highlights the neat property that the inverse of an identity matrix is the identity matrix itself.