Q.Find the value of the following: $\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}$

The key idea is to evaluate a $2 \times 2$ determinant using the formula $\det = ad - bc$.

Step 1: Identify the entries: $a = -1$, $b = 5$, $c = -3$, $d = 2$.

Step 2: Apply the formula:

$\det = (-1)(2) - (5)(-3)$ …

The problem asks for the value of the matrix, but a matrix itself is an array of numbers — it doesn't have a single "value" in the ordinary sense. In the context of determinants (which is the standard interpretation for such a question), we are being asked to compute its determinant. The determinant is a special number associated with a square matrix that tells us, among other things, whether the matrix is invertible and how it scales area (or volume).

For a $2 \times 2$ matrix, the formula is straightforward and worth remembering:

This is the product of the main diagonal entries minus the product of the off-diagonal entries. The "main diagonal" runs from top-left to bottom-right.

Let's apply it step by step.

1. Identify the entries.

   In $\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}$, we have:

   • $a = -1$ (top-left)
   • $b = 5$ (top-right)
   • $c = -3$ (bottom-left)
   • $d = 2$ (bottom-right)

2. Compute $ad$.

   Multiply $a$ and $d$: $(-1) \times 2 = -2$.

3. Compute $bc$.

   Multiply $b$ and $c$: $5 \times (-3) = -15$.

4. Subtract $bc$ from $ad$.

   The determinant is $ad - bc = (-2) - (-15)$. …

Method: Evaluating a $2\times2$ Determinant With Negative Entries

Same baseline $ad-bc$ technique, with extra care for double-negative subtraction.

Steps

Step 1: Identify the four entries with their signs

Label $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ exactly as printed.

Step 2: Compute $ad$ and $bc$ separately

Keep each product's sign explicit before combining them. …

Common Mistakes

Mistake 1: Mishandling the double-negative subtraction $ad-bc$ produces

Why it's wrong: here $ad=-2$ and $bc=-15$, so the formula requires computing $-2-(-15)$ — rushing this step often gives $-2-15=-17$ (treating it as if both were being added negatively) instead of the correct $-2+15=13$. Correct approach: rewrite "subtracting a negative" as an explicit addition ($-2-(-15) = -2+15$) before combining, rather than trying to do it in one mental step.

Mistake 2: Losing track of which entry's sign belongs to $b$ versus $c$ when both diagonal entries are negative …