Q.Find the value of the following: $\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}$

The key idea is that for a $2 \times 2$ matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$, the determinant is $ad - bc$.

Step 1: Identify the entries: $a = 2$, $b = 4$, $c = -5$, $d = -1$.

Step 2: Apply the formula:

Step 3: Simplify:

The determinant is a single number that captures key information about a matrix — for a $2 \times 2$ matrix, it tells you the signed area of the parallelogram formed by its row vectors. The formula itself is simple: multiply the top-left and bottom-right entries, then subtract the product of the top-right and bottom-left entries.

Let’s apply it step by step.

1. Identify the entries.

   In the matrix $\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}$, we have:

   $a = 2$, $b = 4$, $c = -5$, $d = -1$.

2. Compute $ad$.

   $ad = 2 \times (-1) = -2$.

3. Compute $bc$.

   $bc = 4 \times (-5) = -20$.

4. Subtract: $ad - bc$.

   $-2 - (-20) = -2 + 20 = 18$.

Method: Direct ad − bc Evaluation of a 2×2 Determinant

This is the baseline method for evaluating any $2\times2$ determinant with purely numeric entries — the building block every other determinant technique (row operations, cofactor expansion) eventually reduces to.

Steps

Step 1: Identify the four entries in position

For $\begin{vmatrix}a&b\\c&d\end{vmatrix}$, label the entries by position — top-left is $a$, top-right is $b$, bottom-left is $c$, bottom-right is $d$ — reading the matrix exactly as printed, without rearranging anything.

Step 2: Apply the formula

Multiply the main diagonal ($a\times d$) and subtract the product of the anti-diagonal ($b\times c$).

Step 3: Handle negative entries carefully

When any entry is negative, compute each product with its sign first, then carry out the subtraction as adding the opposite — this is where sign errors most often creep in (e.g. subtracting a negative number means adding its absolute value).