To determine the value of [tex]\( a \)[/tex] that makes the matrix
[tex]\[
\begin{pmatrix}
a & 0 \\
0 & a
\end{pmatrix}
\][/tex]
an identity matrix, we first need to understand what an identity matrix is.
An identity matrix is a special type of matrix in which all the elements on the main diagonal are 1, and all other elements are 0. In the context of a 2x2 identity matrix, it is represented as
[tex]\[
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\][/tex]
Comparing this with the given matrix
[tex]\[
\begin{pmatrix}
a & 0 \\
0 & a
\end{pmatrix}
\][/tex]
we observe that for this matrix to be equivalent to the identity matrix, the entries on the main diagonal must both be equal to 1. Therefore, we need both instances of [tex]\( a \)[/tex] in the matrix to be 1.
This means:
[tex]\[
a = 1
\][/tex]
So, the value of [tex]\( a \)[/tex] that makes the given matrix an identity matrix is [tex]\( \boxed{1} \)[/tex].
