Let A be a symmetric matrix of order 2 with integer entries. Then If A is not invertible matrix, then both diagonal entries will be positive only true or false

To assess whether the statement is true or false, we need to analyze the properties of a symmetric matrix \( A \) of order 2 with integer entries that is not invertible. 

Let the matrix \( A \) be given by:
\[
A = \begin{pmatrix} a & b \\ b & d \end{pmatrix}
\]
where \( a, b, d \) are integers.

### Step 1: Condition for Non-Invertibility
A matrix is non-invertible if its determinant is zero. For the matrix \( A \), the determinant is:
\[
\text{det}(A) = ad - b^2
\]
For \( A \) to be non-invertible, the determinant must be zero:
\[
ad - b^2 = 0
\]
This implies:
\[
ad = b^2
\]

### Step 2: Diagonal Entries Analysis
Now, we are asked to check if, under the condition of non-invertibility, the diagonal entries \( a \) and \( d \) will both be positive. 

From the equation \( ad = b^2 \), the product \( ad \) must be a perfect square, specifically \( b^2 \). Since \( b^2 \) is a non-negative integer, \( ad \geq 0 \).

#### Case 1: \( b = 0 \)
If \( b = 0 \), then the equation becomes:
\[
ad = 0
\]
This means that either \( a = 0 \) or \( d = 0 \), or both. In this case, it is possible for either diagonal entry to be zero or negative, so it is not necessary that both diagonal entries will be positive.

#### Case 2: \( b \neq 0 \)
If \( b \neq 0 \), then \( ad = b^2 \) implies that \( ad > 0 \). This means that either both \( a \) and \( d \) are positive, or both are negative (since their product is a positive number).

Thus, in the case where \( b \neq 0 \), it is not guaranteed that both diagonal entries will be positive—they could also both be negative.

### Conclusion
The statement that "if \( A \) is not an invertible matrix, then both diagonal entries will be positive" is **false**. In the case of non-invertibility, the diagonal entries \( a \) and \( d \) could both be positive, both be negative, or one could be zero, depending on the value of \( b \).

### Follow-up Questions:
1. What happens to the diagonal entries if \( b \) is zero?
2. How can you construct a non-invertible symmetric matrix with both diagonal entries being negative?
3. What are the conditions for a symmetric matrix to have both diagonal entries positive?
4. How does the determinant condition impact the possible values of diagonal entries?
5. Could the matrix \( A \) be diagonal with one positive and one negative entry if it's not invertible?

### Tip:
For non-invertible matrices, checking the determinant condition can help you understand the relationship between the diagonal and off-diagonal elements, especially in symmetric matrices.

Explore the properties of non-invertible symmetric matrices with integer entries. Learn about the conditions under which both diagonal entries can be positive or negative, based on the determinant and matrix algebra principles.

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