2x2 行列式計算機：2次正方行列の即時計算と公式

Direct Answer / AI Overview: A 2x2 determinant calculator computes the scalar value of a $2\times 2$ matrix using the fundamental cross-multiplication formula: for matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, $\det(A) = ad - bc$. The absolute value $|\det(A)|$ geometrically represents the area of the parallelogram formed by the matrix’s column vectors in the 2D Cartesian plane. If $\det(A) \neq 0$, the matrix is invertible with inverse $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$.

The $2\times 2$ determinant is the most fundamental building block in linear algebra. It serves as the base case for all higher-order determinant algorithms—including $3\times 3$, $4\times 4$, and $5\times 5$ matrices.

In this guide, we explore the $ad - bc$ formula, its 2D geometric interpretation, matrix inversion rules, Cramer’s rule for solving 2-variable systems, and how to use our free online Determinant Solver.

2x2 Determinant Formula

For any $2\times 2$ square matrix $A$:

$$A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$$

The determinant is calculated as the product of the main diagonal minus the product of the anti-diagonal:

$$\det(A) = |A| = (a \cdot d) - (b \cdot c)$$

     ┌       ┐
     │ a ─── b │   Diagonal Down: a · d
     │   ╳   │
     │ c ─── d │   Diagonal Up:   b · c
     └       ┘
     det(A) = ad - bc

Step-by-Step Worked Examples

Example 1: Standard Positive Matrix

Find $\det(A)$ for $A = \begin{bmatrix} 4 & 7 \ 2 & 6 \end{bmatrix}$:

1. Multiply the main diagonal: $4 \times 6 = 24$
2. Multiply the off-diagonal: $7 \times 2 = 14$
3. Subtract: $\det(A) = 24 - 14 = \mathbf{10}$

Example 2: Matrix with Negative Numbers

Find $\det(B)$ for $B = \begin{bmatrix} 3 & -4 \ -2 & 5 \end{bmatrix}$:

1. Multiply main diagonal: $3 \times 5 = 15$
2. Multiply off-diagonal: $(-4) \times (-2) = +8$
3. Subtract: $\det(B) = 15 - 8 = \mathbf{7}$

Example 3: Zero Determinant (Singular Matrix)

Find $\det(C)$ for $C = \begin{bmatrix} 2 & 6 \ 1 & 3 \end{bmatrix}$:

1. Main diagonal: $2 \times 3 = 6$
2. Off-diagonal: $6 \times 1 = 6$
3. Subtract: $\det(C) = 6 - 6 = \mathbf{0}$

Because $\det(C) = 0$, Row 1 is a linear multiple of Row 2 ($R_1 = 2R_2$). The matrix is singular, has no inverse, and cannot be solved uniquely.

Geometric Meaning: 2D Parallelogram Area

When you plot the two column vectors $\vec{v}_1 = \begin{bmatrix} a \ c \end{bmatrix}$ and $\vec{v}_2 = \begin{bmatrix} b \ d \end{bmatrix}$ on the $(x, y)$ coordinate plane, they form the adjacent sides of a parallelogram.

$$\text{Area of Parallelogram} = |\det(A)| = |ad - bc|$$

• Positive Determinant ($\det > 0$): The vectors preserve standard counterclockwise orientation.
• Negative Determinant ($\det < 0$): The vectors have flipped orientation (reflection across an axis).
• Zero Determinant ($\det = 0$): The two vectors are collinear (parallel), flattening the parallelogram into a 1D line with an area of zero.

Inverting a 2x2 Matrix Using Determinants

A $2\times 2$ matrix $A$ is invertible if and only if $\det(A) \neq 0$. The inverse matrix $A^{-1}$ is calculated directly by swapping diagonal entries, negating off-diagonal entries, and dividing by the determinant:

$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$$

Learn more about matrix inverses in our Matrix Inverse & Adjugate Method Guide.

Solving 2-Variable Systems via Cramer’s Rule

Given the system of linear equations: $$\begin{cases} a_1 x + b_1 y = c_1 \ a_2 x + b_2 y = c_2 \end{cases}$$

Define the coefficient determinants:

• $D = \det\begin{bmatrix} a_1 & b_1 \ a_2 & b_2 \end{bmatrix} = a_1 b_2 - b_1 a_2$
• $D_x = \det\begin{bmatrix} c_1 & b_1 \ c_2 & b_2 \end{bmatrix} = c_1 b_2 - b_1 c_2$
• $D_y = \det\begin{bmatrix} a_1 & c_1 \ a_2 & c_2 \end{bmatrix} = a_1 c_2 - c_1 a_2$

The unique solution is given by: $$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D} \quad (\text{for } D \neq 0)$$

Key Properties of 2x2 Determinants

Frequently Asked Questions (FAQ)

What is the formula for a 2x2 determinant?

The formula for the determinant of a $2\times 2$ matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$ is: $\det(A) = ad - bc$.

Can a 2x2 determinant be negative?

Yes. A negative determinant indicates that the geometric transformation includes a reflection, reversing the spatial orientation of the 2D plane.

How do you calculate a 2x2 determinant online?

You can use our free determinant calculator 2x2 by selecting 2 × 2 matrix size to get instant calculations and inverse verification.