35. If the matrix \( \left(\begin{array}{ccc}6 & k & 2 \\ 1 & -3 & 3 \\ 2 & -k & 2\end{array}\right) \) is not invermbie (i.e. is singular), the value of \( k \) is A - 2 B 3 C 2 d 3

To determine the value of \( k \) for which the matrix \[ A = \begin{pmatrix} 6 & k & 2 \\ 1 & -3 & 3 \\ 2 & -k & 2 \end{pmatrix} \] is singular (not invertible), we need to find when the determinant of the matrix is equal to zero. The determinant of a \( 3 \times 3 \) matrix \[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \] is calculated as: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg). \] For our matrix \( A \): - \( a = 6 \), \( b = k \), \( c = 2 \) - \( d = 1 \), \( e = -3 \), \( f = 3 \) - \( g = 2 \), \( h = -k \), \( i = 2 \) Now, we calculate the determinant: \[ \text{det}(A) = 6((-3)(2) - (3)(-k)) - k(1 \cdot 2 - 3 \cdot 2) + 2(1 \cdot (-k) - (-3) \cdot 2). \] Calculating each term: 1. \( (-3)(2) - (3)(-k) = -6 + 3k = 3k - 6 \) 2. \( 1 \cdot 2 - 3 \cdot 2 = 2 - 6 = -4 \) 3. \( 1 \cdot (-k) - (-3) \cdot 2 = -k + 6 \) Now substituting these back into the determinant formula: \[ \text{det}(A) = 6(3k - 6) - k(-4) + 2(-k + 6). \] Expanding this: \[ = 18k - 36 + 4k + 2(-k + 6). \] Distributing the \( 2 \): \[ = 18k - 36 + 4k - 2k + 12. \] Combining like terms: \[ = (18k + 4k - 2k) + (-36 + 12) = 20k - 24. \] Setting the determinant equal to zero for the matrix to be singular: \[ 20k - 24 = 0. \] Solving for \( k \): \[ 20k = 24 \implies k = \frac{24}{20} = \frac{6}{5}. \] However, this value does not match any of the options provided (A: -2, B: 3, C: 2, D: 3). Let's check if we made any mistakes in the calculations or if the options are incorrect. Revisiting the determinant calculation: 1. \( 6(3k - 6) = 18k - 36 \) 2. \( -k(-4) = 4k \) 3. \( 2(-k + 6) = -2k + 12 \) Combining these: \[ = 18k - 36 + 4k - 2k + 12 = 20k - 24. \] Setting \( 20k - 24 = 0 \) gives \( k = \frac{6}{5} \). Since this does not match any of the provided options, it appears there may be an error in the options or the problem statement. If we check the options again, none of them are correct based on the calculations. Thus, the value of \( k \) for which the matrix is singular is \( k = \frac{6}{5} \), which is not listed among the options.