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

Expand along the first row: multiply each entry by the determinant of the 2x2 matrix left after deleting its row and column, then alternate the signs plus, minus, plus and add. This calculator does it for you.

How do you invert a 3x3 matrix by hand?

Compute the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant. It is nine 2x2 determinants plus a transpose and a division, which is why mistakes are common.

Why is 3x3 such a common size?

It is the size of a rotation matrix in three dimensions, of the inertia tensor in mechanics, and of the homogeneous transformation matrix used for two dimensional graphics. Those three uses cover a large share of practical matrix work.

Can I still use the other operations on this page?

Yes. The page just starts at 3x3. All six operations are available from the tabs, and you can resize either grid at any point.

What if my 3x3 matrix has no inverse?

Then its determinant is zero and it is singular. The calculator detects this and says so rather than returning an incorrect result. Check the determinant tab to confirm.

3x3 Matrix Calculator

About 3x3 Matrix Calculator

The 3x3 matrix is the size where hand calculation stops being comfortable and a calculator starts earning its place. A 2x2 determinant is ad minus bc and takes seconds; a 3x3 determinant expanded along the first row needs three 2x2 determinants and careful sign alternation, and the inverse of a 3x3 by hand requires nine cofactors, a transpose and a division. Both are perfectly doable and both are error prone. This page opens with both grids already set to 3x3 so there is nothing to configure. Three by three is also the size that appears most often in practice outside of pure linear algebra courses, because it is the size of a rotation matrix in three dimensional space, of the inertia tensor in mechanics, and of the transformation matrix for two dimensional homogeneous coordinates used in graphics. All six operations are available from the tabs: addition and subtraction need both matrices at the same size, multiplication needs the inner dimensions to agree, and determinant, inverse and transpose work on matrix A alone. For a larger grid see 4x4 Matrix Calculator.

How to use 3x3 Matrix Calculator

Select an operation from the tab bar (Add, Multiply, etc.)
Set matrix dimensions and enter values in the grid cells
See the result matrix or scalar instantly

Frequently Asked Questions

How do you find the determinant of a 3x3 matrix?: Expand along the first row: multiply each entry by the determinant of the 2x2 matrix left after deleting its row and column, then alternate the signs plus, minus, plus and add. This calculator does it for you.
How do you invert a 3x3 matrix by hand?: Compute the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant. It is nine 2x2 determinants plus a transpose and a division, which is why mistakes are common.
Why is 3x3 such a common size?: It is the size of a rotation matrix in three dimensions, of the inertia tensor in mechanics, and of the homogeneous transformation matrix used for two dimensional graphics. Those three uses cover a large share of practical matrix work.
Can I still use the other operations on this page?: Yes. The page just starts at 3x3. All six operations are available from the tabs, and you can resize either grid at any point.
What if my 3x3 matrix has no inverse?: Then its determinant is zero and it is singular. The calculator detects this and says so rather than returning an incorrect result. Check the determinant tab to confirm.

3x3 Matrix Calculator

About 3x3 Matrix Calculator

How to use 3x3 Matrix Calculator

Frequently Asked Questions

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