Determinants and their Applications

Answer

Step 1: Use the formula for 2×2 determinant: \(ad - bc\) Here, \(a = 6\), \(b = 4\), \(c = 8\), \(d = 2\) Step 2: Compute: \((6)(2) - (8)(4) = 12 - 32 = -20\) Answer: \(-20\)…

Answer

Step 1: Apply 2×2 determinant formula: \((a + b)(a + b) - (2a)(2b)\) Step 2: Expand: \(= a^2 + 2ab + b^2 - 4ab = a^2 - 2ab + b^2\) Step 3: Factor: \(= (a - b)^2\) Answer: \((a - b)^2\)…

Answer

Step 1: Expand determinant: \((x - 3)(x + 3) - x(x + 1) = x^2 - 9 - x^2 - x = -x - 9\) Step 2: Set equal to 6: \(-x - 9 = 6\) Step 3: Solve: \(-x = 15 \Rightarrow x = -15\) Answer: \(x = -15\)…

Answer

Step 1: Expand along R₁: \(1 \cdot \left| \begin{array}{cc} 4 & 1 \\ 2 & 5 \end{array} \right| - 2 \cdot \left| \begin{array}{cc} 2 & 1 \\ 3 & 5 \end{array} \right| + 3 \cdot \left| \begin{array}{cc}…

Answer

Step 1: Expand along C₂: \(-2 \cdot \left| \begin{array}{cc} 3 & 2 \\ 2 & 1 \end{array} \right| + 1 \cdot \left| \begin{array}{cc} 1 & 3 \\ 2 & 1 \end{array} \right| - 3 \cdot \left| \begin{array}{cc…

Answer

Step 1: Minor of \(a_{21} = 5\): Delete R₂, C₁ → \(\left| \begin{array}{cc} 6 & 3 \\ 0 & 8 \end{array} \right| = 48\) Step 2: Minor of \(a_{22} = 2\): Delete R₂, C₂ → \(\left| \begin{array}{cc} 1 & 3…

Answer

Step 1: Use \(C_{ij} = (-1)^{i+j} M_{ij}\) \(C_{21} = (-1)^{3} \cdot 48 = -48\) \(C_{22} = (-1)^{4} \cdot (-13) = -13\) \(C_{23} = (-1)^{5} \cdot (-42) = 42\) Answer: \(C_{21} = -48\), \(C_{22} = …

Answer

Step 1: Find cofactors: \(C_{11} = (+)\left| \begin{array}{cc} 2 & -3 \\ 4 & 5 \end{array} \right| = 10 + 12 = 22\) \(C_{12} = (-)\left| \begin{array}{cc} -6 & -3 \\ 8 & 5 \end{array} \right| = -(-3…