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Matrix Calculator: Determinant, Inverse, Transpose and Product
Type each matrix with one row per line and entries separated by spaces or commas. Choose an operation and get the result, with exact fractions when you enter whole numbers.
Determinant: what it is and how it is found
The determinant is a single number computed from a square matrix. For a 2 by 2 matrix with rows (a, b) and (c, d) it is ad − bc. For larger matrices the tool uses cofactor expansion along the first row: each entry is multiplied by the determinant of the smaller matrix left when its row and column are removed, with alternating plus and minus signs.
For the 3 by 3 matrix with rows (2, −3, 1), (2, 0, −1), (1, 4, 5) the expansion gives 2 × 4 − (−3) × 11 + 1 × 8 = 49. A determinant of zero means the rows are linearly dependent and the matrix is singular, so it has no inverse.
Inverse by cofactors
The inverse of A satisfies A × A−1 = I, the identity matrix. This tool builds it as the transposed cofactor matrix divided by the determinant. For a 2 by 2 matrix with rows (a, b) and (c, d) that is 1 ÷ (ad − bc) times the matrix with rows (d, −b) and (−c, a).
When every entry you type is a whole number, the entries of the inverse are shown as reduced fractions, with a decimal beside them, because the cofactors and the determinant are whole numbers and the division is exact. For decimal inputs the result is rounded to six places. Cofactor expansion is fine for these small sizes; large matrices are normally handled with elimination methods instead.
Multiplication, addition and transpose
Matrix addition and subtraction work entry by entry and require the same size. Multiplication needs the number of columns in A to equal the number of rows in B. Entry (i, j) of the product is the sum of A(i, k) × B(k, j) over k. A 2 by 3 matrix times a 3 by 2 matrix gives a 2 by 2 result.
Order matters: A × B and B × A are usually different, and one may not exist when the other does. The transpose swaps rows and columns, so entry (i, j) moves to (j, i), and a 2 by 3 matrix becomes 3 by 2.
Entering matrices and common errors
- One row per line. Separate entries with spaces, commas or semicolons. Every row needs the same number of entries.
- Size limit. Up to 4 by 4 is accepted, which covers typical coursework.
- Decimals. Use a point, not a comma, as the decimal mark, because commas separate entries.
- Near-singular matrices. Floating-point entries can make a tiny nonzero determinant that is really zero. If an inverse looks absurdly large, check your inputs.
The tool does not do eigenvalues, rank or systems of equations. For solving a linear system, the inverse times the right-hand side gives the solution when the matrix is invertible.
Worked example
Multiply the 2 by 2 matrices [[1, 2], [3, 4]] and [[5, 6], [7, 8]]. Each result entry is a row of A times a column of B.
Inputs
| Operation | Multiply A × B |
|---|---|
| Matrix A | 1 2 3 4 |
| Matrix B | 5 6 7 8 |
Result
Result is 2 by 2.
Product A × B
| Col 1 | Col 2 |
|---|---|
| 19 | 22 |
| 43 | 50 |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
How do I find the determinant of a 3 by 3 matrix?
Expand along the first row: multiply each entry by the determinant of the 2 by 2 matrix left after deleting its row and column, alternating plus and minus signs, and add the results.
When does a matrix have no inverse?
A square matrix has no inverse when its determinant is zero. Such a matrix is called singular, and its rows or columns are linearly dependent.
Why can I not multiply some pairs of matrices?
The number of columns in the first matrix must equal the number of rows in the second. A 2 by 3 matrix cannot be multiplied by another 2 by 3 matrix.
Is matrix multiplication commutative?
No. In general A times B is not equal to B times A, and the reversed product may not even be defined for the given sizes.
Why are some inverse entries shown as fractions?
With whole-number input, each inverse entry is a whole cofactor divided by a whole determinant, so a reduced fraction is exact while a decimal may be rounded.
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References and sources
- Strang, G. Introduction to Linear Algebra, 5th edition, Wellesley-Cambridge Press.