If A is a aquare matrix of order 2 and det. `A=10`, then `((tr. A)^(2)-tr. (A^(2)))` is equal to ______ .

Explanation

To solve the problem, we need to find the value of \(((\text{tr} A)^2 - \text{tr}(A^2))\) given that \(A\) is a square matrix of order 2 and \(\det(A) = 10\).

### Step-by-step Solution:

1. **Define the Matrix**: Let \(A\) be a \(2 \times 2\) matrix represented as:
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

2. **Calculate the Trace of A**: The trace of a matrix \(A\) is defined as the sum of its diagonal elements:
\[ \text{tr}(A) = a + d \]

3. **Calculate the Determinant of A**: The determinant of matrix \(A\) is given by:
\[ \det(A) = ad - bc \]
We know from the problem that \(\det(A) = 10\), so:
\[ ad - bc = 10 \]

4. **Calculate \(A^2\)**: We need to find \(A^2\):
\[ A^2 = A \cdot A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + dc & bc + d^2 \end{pmatrix} \]

5. **Calculate the Trace of \(A^2\)**: The trace of \(A^2\) is:
\[ \text{tr}(A^2) = (a^2 + bc) + (bc + d^2) = a^2 + d^2 + 2bc \]

6. **Substitute into the Expression**: Now we substitute the values into the expression \((\text{tr} A)^2 - \text{tr}(A^2)\):
\[ (\text{tr} A)^2 = (a + d)^2 = a^2 + 2ad + d^2 \]
Therefore,
\[ (\text{tr} A)^2 - \text{tr}(A^2) = (a^2 + 2ad + d^2) - (a^2 + d^2 + 2bc) \]
Simplifying this gives:
\[ = 2ad - 2bc \]

7. **Use the Determinant**: We know that \(ad - bc = 10\). Thus,
\[ 2(ad - bc) = 2 \times 10 = 20 \]

8. **Final Answer**: Therefore, the value of \(((\text{tr} A)^2 - \text{tr}(A^2))\) is:
\[ \boxed{20} \]