Step 1: Concept Definition:
A singular matrix has a determinant of zero. Matrices P, Q, and R are given as singular. This property will be used to determine the unknown variables a, b, and c.
Step 2: Calculation Method:
For a 2x2 matrix \(M = \begin{bmatrix} p & q
r & s \end{bmatrix}\), the determinant is \(|M| = ps - qr\).
A singular matrix M satisfies \(|M| = 0\), thus \(ps - qr = 0\).
Step 3: Detailed Calculations:
Matrix P:
Given \(P = \begin{bmatrix} 2 & 3a
4 & 3 \end{bmatrix}\) is singular, so \(|P| = 0\).
\[ (2)(3) - (3a)(4) = 0 \]\[ 6 - 12a = 0 \]\[ 12a = 6 \]\[ a = \frac{6}{12} = \frac{1}{2} \]Matrix Q:
Given \(Q = \begin{bmatrix} b & 5
2a & 6 \end{bmatrix}\) is singular, so \(|Q| = 0\).
\[ (b)(6) - (5)(2a) = 0 \]\[ 6b - 10a = 0 \]Substituting \(a = 1/2\):
\[ 6b - 10\left(\frac{1}{2}\right) = 0 \]\[ 6b - 5 = 0 \]\[ 6b = 5 \]\[ b = \frac{5}{6} \]Matrix R:
Given \(R = \begin{bmatrix} a^2 + b^2 - c & 1-c
c+1 & c \end{bmatrix}\) is singular, so \(|R| = 0\).
\[ (a^2 + b^2 - c)(c) - (1-c)(c+1) = 0 \]\[ c(a^2 + b^2) - c^2 - (1 - c^2) = 0 \]\[ c(a^2 + b^2) - c^2 - 1 + c^2 = 0 \]\[ c(a^2 + b^2) - 1 = 0 \]\[ c(a^2 + b^2) = 1 \]Substituting the values of a and b:
\[ a^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]\[ b^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \]\[ c\left(\frac{1}{4} + \frac{25}{36}\right) = 1 \]\[ c\left(\frac{9}{36} + \frac{25}{36}\right) = 1 \]\[ c\left(\frac{34}{36}\right) = 1 \]\[ c\left(\frac{17}{18}\right) = 1 \]\[ c = \frac{18}{17} \]Step 4: Final Result:
Calculate the value of \((2a + 6b + 17c)\).
\[ 2a = 2\left(\frac{1}{2}\right) = 1 \]\[ 6b = 6\left(\frac{5}{6}\right) = 5 \]\[ 17c = 17\left(\frac{18}{17}\right) = 18 \]\[ 2a + 6b + 17c = 1 + 5 + 18 = 24 \]The final value is 24.