Matrix calculator, exact answers with steps.

A matrix calculator does the arithmetic of grids of numbers: adding, multiplying, finding a determinant, inverting, row reducing and solving equations. This one takes matrices up to 6 × 6, keeps every answer as an exact fraction, and writes out each step and the formula behind it, so you can check your homework or see where a number came from. Switch to decimals whenever you want.

How to use the matrix calculator

1. Pick the size of Matrix A with the Rows and Columns boxes, then type its numbers. You can type whole numbers, decimals such as 2.5, or fractions such as 3/4. An empty box counts as 0.
2. To paste a table from a spreadsheet, click the top left box and paste. The grid grows to fit, up to 6 × 6.
3. For a two matrix sum, fill Matrix B the same way. Operations that use one matrix grey B out.
4. Choose what to do with the buttons above the grids. The answer and its steps appear under the grids at once.
5. Use Use as A or Use as B to carry an answer into the next calculation, Copy result to paste it back into a spreadsheet, or Copy link to share the matrices.

What each operation does

• A + B and A − B need the same size. Each entry is added or subtracted with the entry in the same place.
• A × B needs the columns of A to equal the rows of B. Entry (i, j) is row i of A times column j of B. A 2 × 3 times a 3 × 4 is a 2 × 4. Order matters: A × B and B × A are usually different, so both buttons are there.
• Determinant, inverse, trace and power need a square matrix. A has an inverse only when its determinant is not 0.
• Rank is the number of independent rows, found as the number of pivots after row reduction.
• Row reduce (RREF) shows every swap, scaling and subtraction until each pivot is 1 and alone in its column.
• Solve Ax = b uses Matrix A for the coefficients and the first column of Matrix B as b. It tells you if there is one solution, none or infinitely many.

Worked example: multiply two 2 × 2 matrices

With A = [1 2; 3 4] and B = [5 6; 7 8], the top left entry is 1·5 + 2·7 = 19. The top right is 1·6 + 2·8 = 22. The bottom row is 3·5 + 4·7 = 43 and 3·6 + 4·8 = 50. So A × B = [19 22; 43 50]. Press B × A and you get [23 34; 31 46], which shows the order matters.

Worked example: determinant and inverse

For A = [1 2; 3 4] the determinant is ad − bc = 1·4 − 2·3 = −2. It is not 0, so A has an inverse: (1 ÷ −2) × [4 −2; −3 1] = [−2 1; 3/2 −1/2]. Multiply A by that and the calculator checks you get the identity matrix. For [2 0 1; 1 3 2; 1 1 1] the determinant is 2(3 − 2) − 0 + 1(1 − 3) = 0, so it is singular and the calculator says it has no inverse instead of giving a wrong answer.

Worked example: solve three equations

Take x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27. Put the coefficients in A as [1 1 1; 0 2 5; 2 5 −1] and the right sides in the first column of B as 6, −4, 27. Choose Solve Ax = b. Row reduction gives x = 5, y = 3 and z = −2. Check the first equation: 5 + 3 − 2 = 6.

Matrix power

Aⁿ means A multiplied by itself n times. For A = [1 1; 1 0], A¹⁰ = [89 55; 55 34], whose entries are Fibonacci numbers. A negative power uses the inverse first, so A⁻² = (A⁻¹)². A power of 0 gives the identity matrix.

Common problems

• Sizes do not match. The message tells you the two sizes and what must agree. For a product, check columns of the first against rows of the second.
• A box is not a number. The message names the row and column. Use a point for decimals, not a comma.
• The inverse does not exist. A singular matrix has determinant 0 and rank below its size. The steps show which row turned into zeros.
• Long decimals. Fractions are exact. Switch Show answers as to Decimals and pick the places you need.

For plain fractions use the fraction calculator, for a single sum the scientific calculator, and to draw a function try the graphing calculator.