Q.Find the value of the following: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

The key idea is that the determinant of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is given by $ad - bc$.

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

Step 2: Apply the formula:

$\det = (1)(4) - (2)(3) = 4 - 6$.

Step 3: Simplify: $4 - 6 = -2$.

The determinant is a single number that captures key information about a matrix: whether it is invertible, the scaling factor of the area it transforms, and more. For a $2 \times 2$ matrix, the formula is straightforward, but it’s worth understanding why it works.

Think of the rows as vectors: $(1, 2)$ and $(3, 4)$. The determinant measures the signed area of the parallelogram they span. The formula $ad - bc$ comes from subtracting the product of the “wrong” diagonal from the product of the “main” diagonal. Here’s how we apply it.

1. Identify the entries. In the matrix $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, we have:

   • $a = 1$ (top-left)
   • $b = 2$ (top-right)
   • $c = 3$ (bottom-left)
   • $d = 4$ (bottom-right)

2. Apply the determinant formula. For any $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is:

Substitute the values:

3. Compute the products. First, $1 \times 4 = 4$. Then, $2 \times 3 = 6$.

4. Subtract. $4 - 6 = -2$.

The result is $-2$. This negative value tells us that the orientation of the vectors is reversed relative to the standard basis — the area is still 2 square units, but with a flipped direction.

Method: Evaluating a $2\times2$ Determinant

The baseline technique used for every $2\times2$ determinant.

Steps

Step 1: Identify the four entries

Label the determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$, reading positions exactly as printed.

Step 2: Apply the formula

Step 3: Compute the two products and subtract

Multiply along the main diagonal (top-left $\times$ bottom-right), multiply along the other diagonal (top-right $\times$ bottom-left), then subtract the second from the first.