Determinant of a 2×2 Matrix Calculator

Published on: October 26 2025
Final Answer: Free • Full Steps: Plus

This Determinant of a 2×2 Matrix Calculator finds the determinant of a \(2 \times 2\) matrix and shows the working line by line. The product along the main diagonal has the product along the other diagonal taken away from it. The result tells you how the matrix scales area, and a determinant of zero means it flattens everything and has no inverse. Following the same pattern every time makes the determinant easy to check and repeat.

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the four entries into the formula.
  4. Multiply the diagonals and subtract.

Formula:

\(\det(A) = ad - bc\)
See Example 1 Hide Example 1

Problem

\(A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the four entries into the formula.
  4. Multiply the diagonals and subtract.

Step 1

Step 1 - Set up the problem.

In this problem: Reading the matrix left to right and top to bottom gives \(a = 4\), \(b = 7\), \(c = 2\) and \(d = 6\).

\(A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: The determinant of a 2 by 2 matrix is the product along the main diagonal minus the product along the other diagonal.

\(\det(A) = ad - bc\)

Step 3

Step 3 - Substitute the four entries into the formula.

In this problem: Replacing the letters with the entries gives \(\det(A) = 4 \cdot 6 - 7 \cdot 2\).

\(\det(A) = 4 \cdot 6 - 7 \cdot 2\)

Step 4

Step 4 - Multiply the diagonals and subtract.

In this problem: The main diagonal gives \(4 \cdot 6 = 24\) and the other diagonal gives \(7 \cdot 2 = 14\), so \(24 - 14 = 10\).

\(\det(A) = 24 - 14 = 10\)

Final Answer

\(\det(A) = 10\)
See Example 2 Hide Example 2

Problem

\(A = \begin{pmatrix} 3 & -2 \\ 5 & 1 \end{pmatrix}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the four entries into the formula.
  4. Multiply the diagonals and subtract.

Step 1

Step 1 - Set up the problem.

In this problem: Reading the matrix left to right and top to bottom gives \(a = 3\), \(b = -2\), \(c = 5\) and \(d = 1\).

\(A = \begin{pmatrix} 3 & -2 \\ 5 & 1 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: The determinant of a 2 by 2 matrix is the product along the main diagonal minus the product along the other diagonal.

\(\det(A) = ad - bc\)

Step 3

Step 3 - Substitute the four entries into the formula.

In this problem: Replacing the letters with the entries gives \(\det(A) = 3 \cdot 1 - (-2) \cdot 5\).

\(\det(A) = 3 \cdot 1 - (-2) \cdot 5\)

Step 4

Step 4 - Multiply the diagonals and subtract.

In this problem: The main diagonal gives \(3 \cdot 1 = 3\) and the other diagonal gives \((-2) \cdot 5 = -10\), so \(3 - (-10) = 13\).

\(\det(A) = 3 - (-10) = 13\)

Final Answer

\(\det(A) = 13\)
Matrix A

Example Problems

Click one to load it into the box and solve it.