Calculators / Linear Algebra / Inverse of a 3×3 Matrix Calculator

Inverse of a 3×3 Matrix Calculator

Published on: October 5 2025

This Inverse of a 3×3 Matrix Calculator helps you find the inverse of a 3×3 matrix and shows each step clearly. It works by finding the determinant, computing the cofactors and adjugate matrix, and then 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

  1. Write the matrix A.
  2. Use the 3×3 determinant formula.
  3. Substitute values to compute det( A ).
  4. Compute the cofactor matrix C and the adjugate adj( A ) = Cᵀ.
  5. Compute A⁻¹ = ( 1 / det( A ) ) · adj( A ).

Determinant formula

det( A ) = a(ei − fh) − b(di − fg) + c(dh − eg),
where A =
a
b
c
d
e
f
g
h
i

Inverse formula

A−1 =
1
det( A )
·
adj( A )
if det( A ) ≠ 0
adj( A ) = CT, where C =
C11
C12
C13
C21
C22
C23
C31
C32
C33

Example 1: 3×3 matrix inverse

Step 1 - Write the matrix A.

In this problem: We start with the given matrix A.

A =
1
2
3
0
1
4
5
6
0

Step 2 - Use the 3×3 determinant formula.

In this problem: This is a standard cofactor expansion formula for a 3×3 matrix.

det( A ) = a(ei − fh) − b(di − fg) + c(dh − eg)
, where A =
a
b
c
d
e
f
g
h
i

Step 3 - Substitute values to compute det( A ).

In this problem: Substitute values to get det( A ) = 1.

det( A ) = ( 1 )×(( 1 )×( 0 ) − ( 4 )×( 6 )) − ( 2 )×(( 0 )×( 0 ) − ( 4 )×( 5 )) + ( 3 )×(( 0 )×( 6 ) − ( 1 )×( 5 )) = 1

Step 4 - Compute the cofactor matrix C and the adjugate adj( A ) = Cᵀ.

In this problem: Compute cofactors, then transpose to get adj( A ).

C =
-24
20
-5
18
-15
4
5
-4
1
adj( A ) = CT =
-24
18
5
20
-15
-4
-5
4
1

Step 5 - Compute A⁻¹ = ( 1 / det( A ) ) · adj( A ).

In this problem: Multiply adj( A ) by ( 1 / det( A ) ) to get A⁻¹.

A−1 =
1
·
-24
18
5
20
-15
-4
-5
4
1
=
-24
18
5
20
-15
-4
-5
4
1

Final answer:

-24
18
5
20
-15
-4
-5
4
1

Example 2: 3×3 matrix inverse

Step 1 - Write the matrix A.

In this problem: We start with the given matrix A.

A =
2
0
1
1
1
0
3
2
1

Step 2 - Use the 3×3 determinant formula.

In this problem: This is a standard cofactor expansion formula for a 3×3 matrix.

det( A ) = a(ei − fh) − b(di − fg) + c(dh − eg)
, where A =
a
b
c
d
e
f
g
h
i

Step 3 - Substitute values to compute det( A ).

In this problem: Substitute values to get det( A ) = 1.

det( A ) = ( 2 )×(( 1 )×( 1 ) − ( 0 )×( 2 )) − ( 0 )×(( 1 )×( 1 ) − ( 0 )×( 3 )) + ( 1 )×(( 1 )×( 2 ) − ( 1 )×( 3 )) = 1

Step 4 - Compute the cofactor matrix C and the adjugate adj( A ) = Cᵀ.

In this problem: Compute cofactors, then transpose to get adj( A ).

C =
1
-1
-1
2
-1
-4
-1
1
2
adj( A ) = CT =
1
2
-1
-1
-1
1
-1
-4
2

Step 5 - Compute A⁻¹ = ( 1 / det( A ) ) · adj( A ).

In this problem: Multiply adj( A ) by ( 1 / det( A ) ) to get A⁻¹.

A−1 =
1
·
1
2
-1
-1
-1
1
-1
-4
2
=
1
2
-1
-1
-1
1
-1
-4
2

Final answer:

1
2
-1
-1
-1
1
-1
-4
2
Matrix A