How to find the inverse of a 3x3 matrix (with a worked example)

How to · Updated 2026-11-10

Check the determinant is not 0, then either row reduce [A | I] or divide the adjugate by det A. Worked example in exact fractions, and how to check it.

A 3x3 matrix A has an inverse only when its determinant is not zero, and then A⁻¹ = adj(A) / det(A), or, the same answer by a different road, you row reduce [A | I] until the left half becomes the identity. The matrix calculator does the row-reduction method in exact fractions and lists every row operation, so you can follow it line by line.

What you need

• The matrix A, all nine entries. Whole numbers, decimals and fractions such as 3/4 are all fine.
• Its determinant. If det A = 0, the matrix is called singular and there is no inverse at all, so it is worth finding this first.
• One of the two hand methods: cofactors and the adjugate (common in school exams), or row reduction (Gauss-Jordan, common at university and much quicker for big matrices).

A few words used below. The identity matrix I has 1s down the main diagonal and 0s everywhere else; multiplying by it changes nothing. The minor of an entry is the 2x2 determinant left when you cover that entry’s row and column. The cofactor is the minor with a + or − sign from the chessboard pattern (+ − + / − + − / + − +). The adjugate is the matrix of cofactors, transposed (rows turned into columns).

Step by step

1. Open the matrix calculator and choose Inverse (with steps) in the Work out list.
2. Type A in the Matrix A box, one row per line, entries separated by spaces or commas. You can also paste [[2, 0, 1], [1, 1, 0], [0, 3, 1]] straight from Python.
3. Read the result. If an inverse exists, the page shows A⁻¹ and det A. It also multiplies A by its answer and tells you when A × A⁻¹ comes out as I exactly. If A is singular, it says so and gives the rank.
4. Follow the steps. Each row operation is written out, such as R2 → R2 − 2R1, with the whole augmented matrix after it.
5. Switch Show numbers as to Decimals if you want the answer as decimals, and use Download result (CSV) to keep it.

The formula for the cofactor route:

A⁻¹ = (1 / det A) × adj(A), where adj(A) = (matrix of cofactors)ᵀ

A worked example

Find the inverse of

A = [2 0 1; 1 1 0; 0 3 1]

(semicolons separate the rows).

1. Determinant, expanding along the first row:

det A = 2(1·1 − 0·3) − 0(1·1 − 0·0) + 1(1·3 − 1·0) = 2 − 0 + 3 = 5

It is not zero, so the inverse exists.

2. Cofactors. Cover each entry’s row and column, take the 2x2 determinant, and apply the sign pattern:

• Row 1: C₁₁ = +(1 − 0) = 1, C₁₂ = −(1 − 0) = −1, C₁₃ = +(3 − 0) = 3
• Row 2: C₂₁ = −(0 − 3) = 3, C₂₂ = +(2 − 0) = 2, C₂₃ = −(6 − 0) = −6
• Row 3: C₃₁ = +(0 − 1) = −1, C₃₂ = −(0 − 1) = 1, C₃₃ = +(2 − 0) = 2

3. Adjugate. Transpose the cofactor matrix, so its rows become columns:

adj(A) = [1 3 −1; −1 2 1; 3 −6 2]

4. Divide by det A = 5:

A⁻¹ = [1/5 3/5 −1/5; −1/5 2/5 1/5; 3/5 −6/5 2/5]

Type the same A into the calculator and it reaches this answer in six steps by row reduction. First it swaps R1 and R2 to get a 1 in the top corner. Then it clears below the pivot with R2 → R2 − 2R1 and scales R2 by −1/2. After that it clears the second column and scales R3 by 2/5. The last step clears the third column. The right half of [A | I] is then exactly the matrix above, and the page confirms det A = 5.

Check your answer

Multiply A by your answer. Every entry of A × A⁻¹ should be 0 except the diagonal, which should be 1. A fast spot check is the first row of A times each column of adj(A), then divide by 5:

• (2, 0, 1)·(1, −1, 3) = 2 + 0 + 3 = 5, so 5/5 = 1
• (2, 0, 1)·(3, 2, −6) = 6 + 0 − 6 = 0
• (2, 0, 1)·(−1, 1, 2) = −2 + 0 + 2 = 0

That gives 1, 0, 0, the first row of I. If one of these is off, the usual culprit is a missed sign in the chessboard pattern or forgetting to transpose.

For a quick picture of what an inverse does, the matrix transform page shows a 2x2 matrix stretching and turning the plane. The inverse is the matrix that undoes it. To keep letters instead of numbers in the entries, use symbolic algebra, because the matrix calculator takes numbers only.

Questions people ask

Why does a zero determinant mean no inverse? det A = 0 means the matrix squashes space flat, losing a dimension. Nothing can stretch it back out, so no matrix undoes it.

Can I use the inverse to solve three simultaneous equations? Yes: if A x = b, then x = A⁻¹ b. In practice, choosing Solve A x = B in the calculator is quicker. It row reduces once and also handles systems with no solution or infinitely many.

Which method should I learn? For a 3x3 in an exam, cofactors are reliable and easy to mark. For 4x4 and up, row reduction needs far fewer sums.