If $\begin{vmatrix} 2 & -5 & 3 \\ x & 1 & 4 \\ 0 & 3 & 2 \end{vmatrix} = 132$, find the value of x.

Analysis

To find the value of $x$, we need to calculate the determinant of the given $3 \times 3$ matrix and equate it to $132$.

Concept: Determinant of a $3 \times 3$ Matrix

For a matrix $A = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$, the determinant is calculated by expanding along the first row:

Step-by-Step Solution

Given the matrix:

1. Expand along the first row: $= 2 \begin{vmatrix} 1 & 4 \\ 3 & 2 \end{vmatrix} - (-5) \begin{vmatrix} x & 4 \\ 0 & 2 \end{vmatrix} + 3 \begin{vmatrix} x & 1 \\ 0 & 3 \end{vmatrix} = 132$

2. Calculate the $2 \times 2$ determinants:

• First part: $2[(1 \times 2) - (4 \times 3)] = 2[2 - 12] = 2[-10] = -20$
• Second part: $+5[(x \times 2) - (4 \times 0)] = 5[2x - 0] = 10x$
• Third part: $+3[(x \times 3) - (1 \times 0)] = 3[3x - 0] = 9x$

3. Combine the results: $-20 + 10x + 9x = 132$

4. Solve for $x$: $19x - 20 = 132$ $19x = 132 + 20$ $19x = 152$ $x = \frac{152}{19}$ $x = 8$

Conclusion

The value of $x$ is 8. Comparing this to the provided options, it matches option B.

Exam Tip

When calculating determinants, always be very careful with the signs of the middle term. Remember the sign pattern for a $3 \times 3$ matrix is:

Since the middle element in the first row was $-5$, the formula $-b$ makes it $-(-5) = +5$. Missing this sign change is the most common mistake students make in JAMB Mathematics!