Question

Given \( M = \begin{bmatrix} 2 & 3 & 7 \\ 6 & 4 & 7 \\ 4 & 6 & 14 \end{bmatrix} \), which of the following statement(s) is/are correct?

• A. The rank of M is 2
• B. The rank of M is 3
• C. The rows of M are linearly independent
• D. The determinant of M is 0

Hint:

To determine the rank of a matrix, perform row reduction or check if its rows or columns are linearly independent. A matrix with a determinant of 0 is singular and has a rank less than its size.

Answer 1

The Correct Option is A, D

Step 1: Calculating the rank of M.
To determine the rank of matrix \( M \), we can reduce it to its row echelon form or calculate its determinant. Since the matrix has 3 rows and 3 columns, we check if all rows are linearly independent or if any row can be expressed as a linear combination of others. By performing row reduction, we find that the rank of \( M \) is 2. Thus, option (A) is correct.
Step 2: Checking the determinant of M.
The determinant of matrix \( M \) is calculated as follows: \[ \text{det}(M) = \begin{vmatrix} 2 & 3 & 7 \\ 6 & 4 & 7 \\ 4 & 6 & 14 \end{vmatrix} \] Expanding this determinant, we get: \[ \text{det}(M) = 2 \begin{vmatrix} 4 & 7 \\ 6 & 14 \end{vmatrix} - 3 \begin{vmatrix} 6 & 7 \\ 4 & 14 \end{vmatrix} + 7 \begin{vmatrix} 6 & 4 \\ 4 & 6 \end{vmatrix} \] After calculating the determinants of the 2x2 matrices, we find that the determinant of \( M \) is 0, confirming that the matrix is singular. Thus, option (D) is correct.
Step 3: Analyzing the linear independence of the rows.
Since the rank of \( M \) is 2, the rows of \( M \) are not linearly independent. Therefore, option (C) is incorrect.
Step 4: Conclusion.
Thus, the correct answers are (A) and (D).