Expand `|(3,x),(x,1)|`

Answer

To expand the determinant \(|(3,x),(x,1)|\), we will follow the steps below:

### Step 1: Write down the determinant
We start with the determinant:
\[ D = \begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} \]

### Step 2: Apply the formula for 2x2 determinants
The formula for the determinant of a 2x2 matrix \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) is given by:
\[ D = ad - bc \]
In our case, \(a = 3\), \(b = x\), \(c = x\), and \(d = 1\).

### Step 3: Substitute the values into the formula
Substituting the values into the determinant formula, we get:
\[ D = (3)(1) - (x)(x) \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ D = 3 - x^2 \]

### Final Answer
Thus, the expanded form of the determinant \(|(3,x),(x,1)|\) is:
\[ 3 - x^2 \]

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