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

Inverse of a 3×3 Matrix Calculator

Published on: October 5 2025

Enter the entries of matrix A (3×3), then compute its inverse A⁻¹.

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
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