non-zero solution whenever \( a+c = 2b \).
Step 1: Construct the Coefficient Matrix
The coefficient matrix \( A \) is:
\[ A = \begin{bmatrix} 1 & 3b & b \\ 1 & 2a & a \\ 1 & 4c & c \end{bmatrix} \]
For a non-zero solution to exist, the determinant of \( A \) must be zero:
\[ \begin{vmatrix} 1 & 3b & b \\ 1 & 2a & a \\ 1 & 4c & c \end{vmatrix} = 0. \]
Step 2: Compute the Determinant
Expanding along the first column:
\[ \begin{vmatrix} 1 & 3b & b \\ 1 & 2a & a \\ 1 & 4c & c \end{vmatrix} = 1 \begin{vmatrix} 2a & a \\ 4c & c \end{vmatrix} - 3b \begin{vmatrix} 1 & a \\ 1 & c \end{vmatrix} + b \begin{vmatrix} 1 & 2a \\ 1 & 4c \end{vmatrix}. \]
Computing Each Minor Determinant
\[ \begin{vmatrix} 2a & a \\ 4c & c \end{vmatrix} = 2ac - 4ac = -2ac. \]
\[ \begin{vmatrix} 1 & a \\ 1 & c \end{vmatrix} = c - a. \]
\[ \begin{vmatrix} 1 & 2a \\ 1 & 4c \end{vmatrix} = 4c - 2a. \]
Substituting:
\[ 1(-2ac) - 3b(c-a) + b(4c - 2a) = 0. \]
\[ -2ac - 3bc + 3ab + 4bc - 2ab = 0. \]
\[ -2ac + ab + bc = 0. \]
Step 3: Conclusion
For a non-trivial solution to exist, we require:
\[ b(a + c) = 2ac. \]
Thus, the correct answer is:
\[ (3) \text{ non-zero solution whenever } b(a+c) = 2ac. \]
The system of simultaneous linear equations :
\[ \begin{array}{rcl} x - 2y + 3z &=& 4 \\ 2x + 3y + z &=& 6 \\ 3x + y - 2z &=& 7 \end{array} \]
Solving the System of Linear Equations
If (x,y,z) = (α,β,γ) is the unique solution of the system of simultaneous linear equations:
3x - 4y + 2z + 7 = 0, 2x + 3y - z = 10, x - 2y - 3z = 3,
then α = ?
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]
Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \).
\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]
\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]