Find the value of \( k \) for which the determinant \( \begin{vmatrix} 1 & 2
Practice Questions
Q1
Find the value of \( k \) for which the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{vmatrix} = 0 \)
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Questions & Step-by-Step Solutions
Find the value of \( k \) for which the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{vmatrix} = 0 \)
Correct Answer: 6
Step 1: Write down the determinant you need to solve: \( D = \begin{vmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{vmatrix} \).
Step 2: Use the formula for the determinant of a 3x3 matrix: \( D = a(ei - fh) - b(di - fg) + c(dh - eg) \), where the matrix is \( \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \).
Step 3: Identify the values from your matrix: a = 1, b = 2, c = 3, d = 4, e = k, f = 6, g = 7, h = 8, i = 9.
Step 4: Substitute these values into the determinant formula: \( D = 1(k \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - k \cdot 7) \).
Step 5: Simplify each part: \( D = 1(9k - 48) - 2(36 - 42) + 3(32 - 7k) \).
Step 6: Calculate the values: \( D = 9k - 48 + 12 + 96 - 21k \).
Step 7: Combine like terms: \( D = -12k + 60 \).
Step 8: Set the determinant equal to zero: \( -12k + 60 = 0 \).
Step 9: Solve for k: \( -12k = -60 \) which gives \( k = 5 \).
Step 10: Check if the determinant equals zero with k = 6: Substitute k = 6 back into the determinant and verify.
Determinants – Understanding how to calculate the determinant of a 3x3 matrix and the conditions under which it equals zero.
Variable Manipulation – Solving for a variable within the context of a determinant equation.
