Question

If the matrix A = \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\), then which of the following is true?

• A. The matrix is invertible
• B. The matrix is singular
• C. The matrix is diagonalizable
• D. The matrix is symmetric

Hint:

For a 3x3 matrix with elements in an arithmetic progression like this one, the determinant is always zero. Notice that the elements in each row (1,2,3), (4,5,6), (7,8,9) and each column (1,4,7), (2,5,8), (3,6,9) are in AP. A property of determinants states that if we perform the operation \(C_2 \rightarrow C_2 - C_1\) and \(C_3 \rightarrow C_3 - C_2\), the new columns will be identical, making the determinant zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a 3x3 matrix A and asked to determine its properties from the given options. The key properties to check are whether the matrix is singular, invertible, or symmetric.

Step 2: Key Formula or Approach (Alternate Method):
Instead of calculating the full determinant, we can check for linear dependence among rows. If rows are linearly dependent, determinant is zero and matrix is singular.

Step 3: Detailed Explanation:
The given matrix is:
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Let's check the relationship between rows:
Row2 - Row1 = [4-1, 5-2, 6-3] = [3, 3, 3]
Row3 - Row2 = [7-4, 8-5, 9-6] = [3, 3, 3]
Since Row2 - Row1 = Row3 - Row2, we have Row1 + Row3 = 2 × Row2.
This means the rows are linearly dependent.
When rows are linearly dependent, the determinant of the matrix is zero.
Since det(A) = 0, the matrix is singular and not invertible.
Now check symmetric property: A^T = [[1,4,7],[2,5,8],[3,6,9]] ≠ A, so not symmetric.

Step 4: Final Answer:
The rows are linearly dependent, so determinant is 0. Therefore, the matrix is singular.