Matrix Calculator
Enter one or two matrices up to 3 × 3 to add, subtract or multiply them, or to find the determinant, inverse or transpose of A. Everything is worked out with exact fractions, so an inverse shows 3/2 rather than 1.4999999, and the working for the determinant or the first entry of a product is written out.
- Result in decimals
- [19, 22; 43, 50]
- Size of the result
- 2 × 2
- Determinant of A
- -2
- Working
- Each entry is a row of A times a column of B: entry (1, 1) = 1 × 5 + 2 × 7 = 19
Result, row by row
| Row | Column 1 | Column 2 | Column 3 |
|---|
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Formula
How to use it
- Choose the operation.
- Type matrix A row by row with a semicolon between rows, such as 1 2; 3 4. Fractions and decimals are fine.
- For A + B, A − B or A × B, type matrix B the same way.
- Read the result in brackets, row by row, or in the table below.
Worked examples
[1 2; 3 4] × [5 6; 7 8]
- Result
- [19, 22; 43, 50]
- Size of the result
- 2 × 2
- Determinant of A
- -2
- Working
- Each entry is a row of A times a column of B: entry (1, 1) = 1 × 5 + 2 × 7 = 19
Inverse of [1 2; 3 4], as exact fractions
- Result
- [-2, 1; 3/2, -1/2]
- Result in decimals
- [-2, 1; 1.5, -0.5]
Determinant of a 3 × 3 matrix by cofactor expansion
- Result
- -59
- Working
- 2(4 × 0 − 1 × 2) − 1(0 × 0 − 1 × 5) + 3(0 × 2 − 4 × 5) = 2 × (-2) − 1 × (-5) + 3 × (-20) = -59
Inverse of a 3 × 3 matrix with determinant 5
- Result
- [1/5, 3/5, -1/5; -1/5, 2/5, 1/5; 3/5, -6/5, 2/5]
- Determinant of A
- 5
Subtracting two 3 × 3 matrices
- Result
- [-8, -6, -4; -2, 0, 2; 4, 6, 8]
- Determinant of A
- 0
Transpose of a 2 × 3 matrix
- Result
- [1, 4; 2, 5; 3, 6]
- Size of the result
- 3 × 2
A matrix with no inverse
- Determinant of A
- 0
- Working
- The determinant of A is 0, so A has no inverse
Rules that trip people up
Matrix multiplication is not commutative: A × B and B × A are usually different, and one of them may not exist at all. A × B needs the number of columns of A to equal the number of rows of B; a 2 × 3 matrix times a 3 × 2 matrix gives a 2 × 2.
Only square matrices have determinants and inverses, and only those with a determinant other than 0 can be inverted. [1 2; 2 4] has determinant 1 × 4 − 2 × 2 = 0, because its second row is twice the first, so it has no inverse.
Checking an inverse
Multiply the matrix by its inverse and you should get the identity matrix, with 1s on the diagonal and 0s everywhere else. For [1 2; 3 4] and its inverse [−2 1; 3/2 −1/2], row 1 times column 1 is 1 × (−2) + 2 × 3/2 = 1, and row 1 times column 2 is 1 × 1 + 2 × (−1/2) = 0.
Questions people ask
What is the determinant of [1 2; 3 4]?
−2. Multiply down each diagonal and subtract: 1 × 4 − 2 × 3 = 4 − 6 = −2.
How do you find the inverse of a 2 × 2 matrix?
Swap a and d, change the signs of b and c, and divide everything by the determinant ad − bc. For [1 2; 3 4] that is [4 −2; −3 1] ÷ (−2) = [−2 1; 1.5 −0.5].
Can you multiply a 2 × 3 matrix by a 3 × 2 matrix?
Yes. The inner sizes match (3 and 3), and the answer takes the outer sizes, so it is 2 × 2.
Why does my matrix have no inverse?
Its determinant is 0, which happens when one row or column can be made from the others. Such a matrix is called singular.