Final answer:
To determine the value of k for which the matrix A has rank 2, the determinant of the 3x3 matrix must be zero. By calculating the determinant and setting it to zero, we find k = 5.2.
Step-by-step explanation:
To find the value of k for which the matrix A has rank 2, we need to look at the determinants of the submatrices of A. For a 3x3 matrix to have rank 2, at least one of the 2x2 submatrices must have a non-zero determinant (which makes its rank at least 2), and the determinant of the entire 3x3 matrix must be zero (which prevents its rank from being 3).
Let's consider the submatrices that include the third row of A because they're the ones that include k:
- Submatrix excluding the first column: det | 4 6 | = 4(-3) - 6*2 = -12 - 12 = -24 (Non-zero, so the rank is at least 2)
- Submatrix excluding the second column: det | 7 6 | = 7(k) - 6*3 = 7k - 18
- Submatrix excluding the third column: det | 7 4 | = 7(-3) - 4*3 = -21 - 12 = -33 (Non-zero, so the rank is at least 2)
To ensure the whole matrix has rank 2, the determinant of the entire matrix must equal zero:
det(A) = 7(-3k + 18) - 4(6k - 18) + 6(6-0)
Which simplifies to:
-21k + 126 -24k +72 + 36 = 0
Combining like terms gives:
-45k + 234 = 0
k = 234 / 45
k = 5.2