Q.Evaluate $\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}$.

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

Step 2: Apply the formula:

$\text{Determinant} = (2)(2) - (4)(-1)$.

Step 3: Simplify: $4 - (-4) = 4 + 4 = 8$.

Why Determinants Matter

A determinant is a single number that captures essential information about a square matrix. For a $2\times 2$ matrix, it tells you whether the matrix is invertible (non-zero determinant) or singular (zero determinant). Geometrically, it represents the area scaling factor of the linear transformation the matrix describes — a negative determinant means the transformation flips orientation.

The formula itself comes from solving a system of two linear equations. If you have:

then eliminating variables leads to the expression $ad - bc$ appearing in the denominator. That's why it's so fundamental.

Step-by-Step Evaluation

1. Identify the entries. For a $2\times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is defined as:

In our matrix $\begin{pmatrix} 2 & 4 \\ -1 & 2 \end{pmatrix}$, we have:

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

2. Apply the formula. Substitute into $ad - bc$:

3. Compute carefully — watch the signs. First term: $2 \times 2 = 4$ Second term: $4 \times (-1) = -4$, but note the minus sign in front: $- (4)(-1) = -(-4) = +4$ So:

Method: Direct Evaluation of a 2×2 Determinant

This is the base method every larger determinant technique eventually reduces to — evaluating a $2\times2$ determinant directly from its formula.

Steps

Step 1: Identify the four entries

For $\begin{vmatrix}a&b\\c&d\end{vmatrix}$, read off $a$ (top-left), $b$ (top-right), $c$ (bottom-left), $d$ (bottom-right) exactly as printed, including their signs.

Step 2: Apply the formula

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

Step 3: Handle negative entries carefully

When any entry is negative, bracket it before multiplying — e.g. write $b\times c$ as $(4)\times(-1)$, not left ambiguous mid-calculation, so the final subtraction's sign is unambiguous.

Step 4: Simplify

Combine the two terms, watching the double-negative that appears when $bc$ is itself negative (subtracting a negative becomes addition).

This direct formula is the fastest route for any $2\times2$ determinant with pure numeric entries — no row operations are needed at this size.