Determine the value of \( k \) such that \( \begin{vmatrix} 1 & 2 & 3 \\

Determine the value of \( k \) such that \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & k \end{vmatrix} = 0 \).

Determine the value of \( k \) such that \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & k \end{vmatrix} = 0 \).

Step 1: Write down the determinant you need to solve: | 1 2 3 |
Step 2: Write down the second row of the determinant: | 4 5 6 |
Step 3: Write down the third row of the determinant, which includes k: | 7 8 k |
Step 4: Use the formula for a 3x3 determinant: |A| = a(ei - fh) - b(di - fg) + c(dh - eg), where A is the matrix and a, b, c, d, e, f, g, h, i are the elements of the matrix.
Step 5: Identify the elements from the matrix: a = 1, b = 2, c = 3, d = 4, e = 5, f = 6, g = 7, h = 8, i = k.
Step 6: Substitute these values into the determinant formula: |A| = 1(5k - 48) - 2(4k - 42) + 3(32 - 35).
Step 7: Simplify the expression: |A| = 5k - 48 - 8k + 84 - 9.
Step 8: Combine like terms: |A| = -3k + 27.
Step 9: Set the determinant equal to zero: -3k + 27 = 0.
Step 10: Solve for k: -3k = -27, so k = 9.

Determinants – Understanding how to calculate the determinant of a 3x3 matrix and the conditions under which it equals zero.
Linear Dependence – Recognizing that a determinant of zero indicates that the rows (or columns) of the matrix are linearly dependent.