Inverse of a 2×2 Matrix Calculator

Published on: September 28 2025

This Inverse of a 2×2 Matrix Calculator helps you find the inverse of a 2×2 matrix and shows each step clearly. It works by first finding the determinant, then swapping the main diagonal entries, changing the signs of the off-diagonal entries, and multiplying by the reciprocal of the determinant. This makes it useful for checking answers, understanding how a matrix inverse is found, and practising linear algebra step by step.

Step-by-step method

Identify the entries a, b, c, and d in the 2×2 matrix.
Find the determinant using det( A ) = ad − bc, then check that it is not 0.
Use the inverse matrix pattern shown in the formula.
Multiply the new matrix by 1 / det( A ) and simplify.

Determinant formula

det( A ) = ad − bc,

where A =

a	b
c	d

Inverse formula

A⁻¹ =

det( A )

d	−b
−c	a

if det( A ) 0

Example 1: Take the matrix below.

A =

2	1
5	3

Step 1 - Identify the entries a, b, c, and d in the 2×2 matrix.

In this problem: For a 2×2 matrix, a is top-left, b is top-right, c is bottom-left, and d is bottom-right.

a = 2, b = 1, c = 5, d = 3

Step 2 - Find the determinant using det( A ) = ad − bc, then check that it is not 0.

In this problem: The determinant is 1. Since 1 0, the matrix has an inverse.

det( A ) = ( 2 )×( 3 ) − ( 1 )×( 5 ) = 1,

1 0

Step 3 - Use the inverse matrix pattern shown in the formula.

In this problem: Use the inverse matrix pattern from the formula, then substitute this problem’s values.

d	−b
−c	a

3	-1
-5	2

Step 4 - Multiply the new matrix by 1 / det( A ) and simplify.

In this problem: Multiply each entry by the reciprocal of the determinant and simplify.

A⁻¹ =

3	-1
-5	2

3	-1
-5	2

Final answer:

3	-1
-5	2

Example 2: Take the matrix below.

A =

4	-2
1	1

Step 1 - Identify the entries a, b, c, and d in the 2×2 matrix.

In this problem: For a 2×2 matrix, a is top-left, b is top-right, c is bottom-left, and d is bottom-right.

a = 4, b = -2, c = 1, d = 1

Step 2 - Find the determinant using det( A ) = ad − bc, then check that it is not 0.

In this problem: The determinant is 6. Since 6 0, the matrix has an inverse.

det( A ) = ( 4 )×( 1 ) − ( -2 )×( 1 ) = 6,

6 0

Step 3 - Use the inverse matrix pattern shown in the formula.

In this problem: Use the inverse matrix pattern from the formula, then substitute this problem’s values.

d	−b
−c	a

1	2
-1	4

Step 4 - Multiply the new matrix by 1 / det( A ) and simplify.

In this problem: Multiply each entry by the reciprocal of the determinant and simplify.

A⁻¹ =

1	2
-1	4

-1

Final answer:

-1

Matrix A