Find four matrices M in the form (
a b
b a
) such that M2 = M + I where I = (
1 0
0 1
).

Given that \( M = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \) and the condition \( M^2 = M + I \), where \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), let's find the values of \( a \) and \( b \) that satisfy this equation.

### Step 1: Calculate \( M^2 \)
\[ M^2 = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \times \begin{pmatrix} a & b \\ b & a \end{pmatrix} \]
\[ M^2 = \begin{pmatrix} a^2 + b^2 & ab + ab \\ ab + ab & b^2 + a^2 \end{pmatrix} = \begin{pmatrix} a^2 + b^2 & 2ab \\ 2ab & a^2 + b^2 \end{pmatrix} \]

### Step 2: Substitute into the equation \( M^2 = M + I \)
We have:
\[ \begin{pmatrix} a^2

Explore solutions to finding matrices M in the form (a b; b a) such that M^2 = M + I, where I is the identity matrix. Learn the application of matrix algebra and quadratic equations in solving this problem. Suitable for college-level mathematics.

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