Q.Find the value of the following: $\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}$

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

Step 1: Identify the entries: $a = 2$, $b = -2$, $c = 4$, $d = 3$.

Step 2: Apply the formula: …

The determinant is a single number that captures key information about a matrix — for a $2\times 2$ matrix, it tells you the area scaling factor of the linear transformation it represents. The formula is straightforward: multiply the top-left and bottom-right entries, then subtract the product of the top-right and bottom-left entries.

Let’s work through it:

1. Identify the entries.

   For the matrix $\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}$, we have:

   • $a = 2$ (top-left)
   • $b = -2$ (top-right)
   • $c = 4$ (bottom-left)
   • $d = 3$ (bottom-right)

2. Apply the determinant formula.

   The determinant is $ad - bc$. Substituting:

3. Compute carefully. First term: $2 \times 3 = 6$. Second term: $(-2) \times 4 = -8$. But note the minus sign in front: $ad - bc$ means we subtract $bc$, so: $\det = 6 - (-8) = 6 + 8 = 14$ …

Method: Evaluating a $2\times2$ Determinant

The baseline technique for any $2\times2$ determinant with mixed positive/negative entries.

Steps

Step 1: Identify the four entries, keeping their signs

Label $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$, reading each entry exactly as printed, including its sign.

Step 2: Apply the formula

Step 3: Compute each product carefully, then subtract …