Consider the following statements :
1. The matrix `({:(1,2,1),(a,2a,1),(b,2b,1):})`is singular.
2. The matrix `({:(c,2c,1),(a,2a,1),(b,2b,1):})`is non-singular.

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To determine whether the given matrices are singular or non-singular, we need to calculate their determinants. A matrix is singular if its determinant is zero; otherwise, it is non-singular.

### Step 1: Analyze the first matrix
The first matrix is:
\[ A = \begin{pmatrix} 1 & 2 & 1 \\ a & 2a & 1 \\ b & 2b & 1 \end{pmatrix} \]

### Step 2: Calculate the determinant of matrix A
Using the determinant formula for a 3x3 matrix:
\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 2a & 1 \\ 2b & 1 \end{vmatrix} - 2 \cdot \begin{vmatrix} a & 1 \\ b & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} a & 2a \\ b & 2b \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \(\begin{vmatrix} 2a & 1 \\ 2b & 1 \end{vmatrix} = 2a \cdot 1 - 1 \cdot 2b = 2a - 2b\)
2. \(\begin{vmatrix} a & 1 \\ b & 1 \end{vmatrix} = a \cdot 1 - 1 \cdot b = a - b\)
3. \(\begin{vmatrix} a & 2a \\ b & 2b \end{vmatrix} = a \cdot 2b - 2a \cdot b = 2ab - 2ab = 0\)

Now substituting back into the determinant formula:
\[ \text{det}(A) = 1(2a - 2b) - 2(a - b) + 1(0) \]
\[ = 2a - 2b - 2a + 2b = 0 \]

### Conclusion for the first matrix
Since \(\text{det}(A) = 0\), the first matrix is singular.

### Step 3: Analyze the second matrix
The second matrix is:
\[ B = \begin{pmatrix} c & 2c & 1 \\ a & 2a & 1 \\ b & 2b & 1 \end{pmatrix} \]

### Step 4: Calculate the determinant of matrix B
Using the same determinant formula:
\[ \text{det}(B) = c \cdot \begin{vmatrix} 2a & 1 \\ 2b & 1 \end{vmatrix} - 2c \cdot \begin{vmatrix} a & 1 \\ b & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} a & 2a \\ b & 2b \end{vmatrix} \]

Calculating the 2x2 determinants (same as before):
1. \(\begin{vmatrix} 2a & 1 \\ 2b & 1 \end{vmatrix} = 2a - 2b\)
2. \(\begin{vmatrix} a & 1 \\ b & 1 \end{vmatrix} = a - b\)
3. \(\begin{vmatrix} a & 2a \\ b & 2b \end{vmatrix} = 0\)

Substituting back into the determinant formula:
\[ \text{det}(B) = c(2a - 2b) - 2c(a - b) + 0 \]
\[ = c(2a - 2b - 2a + 2b) = c(0) = 0 \]

### Conclusion for the second matrix
Since \(\text{det}(B) = 0\), the second matrix is also singular.

### Final Result
1. The first statement is **correct**: The matrix \(A\) is singular.
2. The second statement is **incorrect**: The matrix \(B\) is also singular.