Consider a 2 × 2 symmetric matrix  show that  is indefinite if and only if det   0

To prove that a \( 2 \times 2 \) symmetric matrix \(\mathbf{A}\) is indefinite if and only if \(\det(\mathbf{A}) < 0\), let's start by defining what it means for a matrix to be indefinite and by considering the general form of a \( 2 \times 2 \) symmetric matrix.

### Step 1: General Form of a Symmetric Matrix
A \( 2 \times 2 \) symmetric matrix \(\mathbf{A}\) has the form:
\[
\mathbf{A} = \begin{pmatrix} 
a & b \\
b & c 
\end{pmatrix}
\]
where \( a \), \( b \), and \( c \) are real numbers.

### Step 2: Eigenvalues and Indefiniteness
A matrix \(\mathbf{A}\) is **indefinite** if it has both positive and negative eigenvalues. For a \( 2 \times 2 \) matrix, the eigenvalues \(\lambda_1\) and \(\lambda_2\) are the roots of the characteristic equation:
\[
\det(\mathbf{A} - \lambda \mathbf{I}) = 0
\]
where \(\mathbf{I}\) is the identity matrix. Expanding the determinant, we get:
\[
\det\left(\begin{pmatrix} 
a & b \\
b & c 
\end{pmatrix} - \lambda \begin{pmatrix} 
1 & 0 \\
0 & 1 
\end{pmatrix}\right) = \det\begin{pmatrix} 
a - \lambda & b \\
b & c - \lambda 
\end{pmatrix} = (a - \lambda)(c - \lambda) - b^2
\]
This simplifies to the quadratic equation:
\[
\lambda^2 - (a+c)\lambda + (ac - b^2) = 0
\]
The eigenvalues \(\lambda_1\) and \(\lambda_2\) are the solutions of this quadratic equation.

### Step 3: Discriminant Analysis
For the matrix to be indefinite, the eigenvalues must have opposite signs. The discriminant of the quadratic equation is:
\[
\Delta = (a+c)^2 - 4(ac - b^2)
\]
For real eigenvalues (which is the case for symmetric matrices), the discriminant \(\Delta\) must be non-negative. Specifically, if one eigenvalue is positive and the other is negative, the product of the eigenvalues (which is \(\det(\mathbf{A})\)) must be negative.

### Step 4: Determinant Condition
The determinant of the matrix \(\mathbf{A}\) is given

Explore how to prove that a 2x2 symmetric matrix is indefinite using determinants. Understand the relationship between eigenvalues, quadratic forms, and the discriminant. Suitable for advanced high school students and learners of linear algebra.

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