How do you find the determinant of a 3x3 matrix?

Expand along the first row using cofactor expansion. Multiply each top-row entry by the determinant of its 2x2 minor, apply the alternating signs + - +, and add the three results. For [ a b c ; d e f ; g h i ], det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).

Why doesn't my matrix have an inverse?

A matrix has no inverse when its determinant equals zero - such a matrix is called singular. Because finding an inverse requires dividing by the determinant, a determinant of 0 makes the inverse undefined. It usually means one row is a multiple or combination of the others.

What is the difference between the transpose and the inverse of a matrix?

The transpose just swaps rows and columns and always exists. The inverse is the matrix that undoes multiplication, so A times its inverse equals the identity matrix, and it only exists when the determinant is non-zero. They are different operations and normally produce different results.

What size matrices does this calculator support?

This matrix calculator works with 2x2 and 3x3 matrices. It computes the determinant, inverse, and trace for square matrices of those sizes, and the transpose for either size.

What is the trace of a matrix used for?

The trace is the sum of the main-diagonal entries of a square matrix. It is a quick summary value that stays the same when a matrix is transposed, and it appears throughout linear algebra as a simple, fast-to-compute property of a matrix.

Matrix Calculator

Formula

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Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

Use this free matrix calculator to find the determinant, inverse, transpose, or trace of any 2×2 or 3×3 matrix, with every step shown. Enter your numbers, choose an operation, and read the worked solution — no login, no cost. The guide below teaches the by-hand method for each operation so you can reproduce the answer on homework and exams.

Reviewed by a math educator · Last updated September 2026 · Methods used: ad − bc for 2×2, cofactor expansion for the 3×3 determinant, and the adjugate method for the inverse.

What This Matrix Calculator Does

A matrix is a rectangular grid of numbers arranged in rows and columns. This tool focuses on the four operations you need most for an Algebra 2 or first-year linear algebra problem, applied to small square matrices:

Determinant — a single number computed from a square matrix that tells you, among other things, whether the matrix can be inverted.
Inverse — the matrix that "undoes" the original; multiplying a matrix by its inverse gives the identity matrix.
Transpose — the matrix you get by swapping rows and columns (flipping across the main diagonal).
Trace — the sum of the entries on the main diagonal (top-left to bottom-right) of a square matrix.

Supported sizes: 2×2 and 3×3 matrices only. Determinant, inverse, and trace require a square matrix (same number of rows and columns); the transpose works on either supported size.

How to Use the Calculator

Computing a result takes four steps. The tool shows the full step-by-step solution alongside each answer.

Pick the size. Choose a 2×2 or 3×3 grid to match your problem.
Enter the matrix values. Type a number into each cell. Whole numbers, decimals, and negatives are all accepted.
Choose an operation. Select determinant, inverse, transpose, or trace.
Read the step-by-step solution. Press Calculate to see the answer, a visual, and every intermediate step. Want to try first? Use the Hint button before revealing the full solution.

Determinant of a Matrix (2×2 and 3×3)

The determinant is a single number you can calculate from any square matrix. It tells you whether the matrix is invertible: a determinant of zero means no inverse exists (see below). The determinant is written as det(A) or with vertical bars, |A|.

2×2 Determinant — the ad − bc Rule

For a 2×2 matrix with entries a, b, c, d, multiply the main diagonal and subtract the product of the other diagonal:

A = [ a b ; c d ] → det(A) = ad − bc

Worked example. Find the determinant of [ 4 7 ; 2 6 ]:

Multiply the main diagonal. a × d = 4 × 6 = 24.
Multiply the anti-diagonal. b × c = 7 × 2 = 14.
Subtract. det(A) = 24 − 14 = 10.

3×3 Determinant — Cofactor Expansion

For a 3×3 matrix, expand along the first row. Each top-row entry is multiplied by the determinant of the 2×2 minor left over when you cross out that entry's row and column, and the signs alternate + − +:

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

using the labelling [ a b c ; d e f ; g h i ]. Worked example. Find the determinant of [ 1 2 3 ; 4 5 6 ; 7 8 10 ]:

Expand on the first entry (a = 1). Cross out row 1 and column 1, leaving the minor [ 5 6 ; 8 10 ]. Its determinant is 5×10 − 6×8 = 50 − 48 = 2. Contribution: +1 × 2 = 2.
Expand on the second entry (b = 2), with a minus sign. The minor is [ 4 6 ; 7 10 ], determinant 4×10 − 6×7 = 40 − 42 = −2. Contribution: −2 × (−2) = +4.
Expand on the third entry (c = 3), with a plus sign. The minor is [ 4 5 ; 7 8 ], determinant 4×8 − 5×7 = 32 − 35 = −3. Contribution: +3 × (−3) = −9.
Add the three contributions. det(A) = 2 + 4 − 9 = −3.

Tip: The alternating sign pattern for the first row is +, −, +. Forgetting the middle minus sign is the single most common 3×3 determinant error.

Inverse of a Matrix

The inverse of a square matrix A, written A⁻¹, is the matrix that satisfies A × A⁻¹ = I, where I is the identity matrix (1s on the diagonal, 0s elsewhere). Only square matrices with a non-zero determinant have an inverse. The inverse is what lets a systems of equations calculator solve a linear system in one move.

2×2 Inverse — the Swap-and-Negate Shortcut

For a 2×2 matrix, swap a and d, negate b and c, then divide every entry by the determinant:

A⁻¹ = (1 / (ad − bc)) × [ d −b ; −c a ]

Worked example. Invert [ 4 7 ; 2 6 ]:

Find the determinant. ad − bc = 4×6 − 7×2 = 24 − 14 = 10. It is non-zero, so an inverse exists.
Swap a and d; negate b and c. The adjugate is [ 6 −7 ; −2 4 ].
Divide every entry by the determinant (10). A⁻¹ = [ 0.6 −0.7 ; −0.2 0.4 ].
Check. Multiply A × A⁻¹; the result is the identity matrix [ 1 0 ; 0 1 ], confirming the inverse.

3×3 Inverse — the Adjugate Method

For a 3×3 matrix the calculator uses the adjugate method, which builds on the determinant section above:

Compute the determinant by cofactor expansion. If it is zero, stop — there is no inverse.
Build the matrix of cofactors. For each entry, find the 2×2 minor determinant and apply the checkerboard sign pattern (+ − + / − + − / + − +).
Transpose the cofactor matrix to get the adjugate (this is the same row↔column swap described in the transpose section).
Divide the adjugate by the determinant. The result is A⁻¹.

When a Matrix Has No Inverse

A matrix has no inverse exactly when its determinant equals zero. Such a matrix is called singular (a matrix that does have an inverse is invertible or non-singular).

The reason is built into the formula: every inverse requires dividing by the determinant, and division by zero is undefined. A zero determinant means the rows (or columns) are linearly dependent — one row is a multiple of another, or one is a combination of the others — so the matrix collapses information that an inverse would need to recover.

Worth knowing: If you ask this tool for an inverse and get a "no inverse" result, that is not an input error — it means your matrix is singular because its determinant is 0.

Transpose of a Matrix

The transpose of a matrix, written Aᵀ, is what you get by swapping its rows and columns: the first row becomes the first column, the second row becomes the second column, and so on. Entry (row i, column j) moves to position (row j, column i).

[ 1 2 3 ; 4 5 6 ] → [ 1 4 ; 2 5 ; 3 6 ]

Here a 2×3 matrix becomes a 3×2 matrix — the dimensions flip. The transpose is the only operation on this page that does not require a square matrix.

Transpose of a transpose: applying the transpose twice returns the original matrix, so (Aᵀ)ᵀ = A.
Symmetric matrices: if a square matrix is unchanged by transposing (A = Aᵀ), it is called symmetric — its entries mirror across the main diagonal.

Trace of a Matrix

The trace of a square matrix, written tr(A), is simply the sum of the entries on the main diagonal — the line running from the top-left corner to the bottom-right corner. Off-diagonal entries are ignored.

tr([ a b c ; d e f ; g h i ]) = a + e + i

Worked example. For [ 2 9 1 ; 4 5 7 ; 0 3 8 ], the diagonal entries are 2, 5, and 8, so tr(A) = 2 + 5 + 8 = 15. Like the determinant and inverse, the trace is only defined for square matrices.

Worked Example — One 3×3 Matrix Through All Four Operations

To see how the operations connect, run the single matrix A = [ 2 0 1 ; 3 1 2 ; 1 1 0 ] through each one:

Determinant. Expanding along the first row: 2(1×0 − 2×1) − 0(…) + 1(3×1 − 1×1) = 2(−2) − 0 + 1(2) = −4 + 2 = −2.
Inverse. The determinant is −2 (non-zero), so an inverse exists. Build the cofactor matrix, transpose it to the adjugate, then divide by −2 to get A⁻¹.
Transpose. Swap rows and columns: Aᵀ = [ 2 3 1 ; 0 1 1 ; 1 2 0 ].
Trace. Add the main diagonal of A: 2 + 1 + 0 = 3.

The connection: The determinant decided whether an inverse existed, the transpose appeared as a step inside the inverse, and the trace read straight off the same diagonal — four operations, one matrix.

Common Mistakes to Avoid

Mistake 1 — Forgetting the middle minus in a 3×3 determinantThe first-row cofactor signs alternate + − +. The middle term is always subtracted; dropping that minus is the most common 3×3 error.

Mistake 2 — Swapping the wrong entries in a 2×2 inverseSwap a and d (the main diagonal), and negate b and c (the off-diagonal). Reversing this gives the wrong matrix.

Mistake 3 — Forgetting to divide by the determinantThe adjugate (swapped/negated matrix) is not the inverse on its own — you must divide every entry by the determinant to finish.

Mistake 4 — Confusing the transpose with the inverseThe transpose only swaps rows and columns; the inverse undoes multiplication. They are different operations and usually give different matrices.

Mistake 5 — Trying to invert or trace a non-square matrixDeterminant, inverse, and trace require equal rows and columns. Only the transpose works on a non-square matrix.

Frequently Asked Questions

How do you find the determinant of a 3×3 matrix?

Expand along the first row using cofactor expansion. Multiply each top-row entry by the determinant of its 2×2 minor, apply the alternating signs + − +, and add the three results. For [ a b c ; d e f ; g h i ], det(A) = a(ei − fh) − b(di − fg) + c(dh − eg).
Why doesn't my matrix have an inverse?

A matrix has no inverse when its determinant equals zero — such a matrix is called singular. Because finding an inverse requires dividing by the determinant, a determinant of 0 makes the inverse undefined. It usually means one row is a multiple or combination of the others.
What is the difference between the transpose and the inverse of a matrix?

The transpose (Aᵀ) just swaps rows and columns and always exists. The inverse (A⁻¹) is the matrix that undoes multiplication, so A × A⁻¹ equals the identity matrix, and it only exists when the determinant is non-zero. They are different operations and normally produce different results.
What size matrices does this calculator support?

This matrix calculator works with 2×2 and 3×3 matrices. It computes the determinant, inverse, and trace for square matrices of those sizes, and the transpose for either size.
What is the trace of a matrix used for?

The trace is the sum of the main-diagonal entries of a square matrix. It is a quick summary value that stays the same when a matrix is transposed, and it appears throughout linear algebra as a simple, fast-to-compute property of a matrix.