2x2 Matrix Determinant

What is a 2x2 Matrix Determinant?

The determinant of a 2x2 matrix is a scalar value that provides important information about the matrix's properties. It is used in various applications of linear algebra, including solving systems of linear equations, finding inverse matrices, and determining linear transformations.

The Formula

For a 2x2 matrix A:

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

The determinant |A| is calculated as:

\[|A| = ad - bc\]

Where:

• a is the element in the first row, first column
• b is the element in the first row, second column
• c is the element in the second row, first column
• d is the element in the second row, second column

Calculation Steps

1. Multiply the elements on the main diagonal (top-left to bottom-right): ad
2. Multiply the elements on the other diagonal (top-right to bottom-left): bc
3. Subtract the second result from the first: ad - bc

Example

Let's calculate the determinant of this matrix:

\[A = \begin{bmatrix} 3 & 8 \\ 4 & 6 \end{bmatrix}\]

Step-by-step calculation:

1. \(ad = 3 \times 6 = 18\)
2. \(bc = 8 \times 4 = 32\)
3. \(|A| = ad - bc = 18 - 32 = -14\)

The determinant is -14.

Visual Representation

This diagram illustrates the calculation of the determinant for the 2x2 matrix A, resulting in -14.