Calculator

Matrix Calculator with Steps

Multiply matrices, take determinants and inverses, and row reduce to reduced echelon form, with every elementary row operation written in the notation your marker expects.

Solve a matrix problem now

First solution free

How to use it

Enter matrices in bracket notation, one row at a time: [[2,1],[5,3]]. Name the operation you want — multiply, determinant, inverse, transpose, rank, or row reduce — because several of them apply to the same input. For a system of equations written as an augmented matrix, say so, and the working keeps the bar in place. Systems given as ordinary equations are usually clearer through the system of equations solver.

The method behind it

A matrix is a rectangular array of numbers, and its shape governs what you are allowed to do with it. Addition needs identical dimensions. Product $A B$ needs the number of columns of $A$ to equal the number of rows of $B$ , and the result takes the outer dimensions.

(A B)_{ij} = k = 1 \sum n a_{ik} b_{k j}

Each entry of the product is a row of the left matrix paired term by term with a column of the right matrix. That asymmetry is why $A B$ and $B A$ are usually different matrices and occasionally not even the same size.

The determinant of a square matrix measures how much the transformation scales area, or volume in three dimensions. For a two-by-two matrix it is a single product difference.

det [a c b d] = a d - b c

For three-by-three, expand along any row or column, multiplying each entry by the determinant of the two-by-two matrix left after deleting that entry row and column, with signs alternating from a plus in the top-left corner. Choosing a row containing a zero removes a term entirely.

det 101240356 = 1 (4 \cdot 6 - 5 \cdot 0) - 2 (0 \cdot 6 - 5 \cdot 1) + 3 (0 \cdot 0 - 4 \cdot 1) = 24 + 10 - 12 = 22

A zero determinant means the transformation collapses space, and a collapse cannot be undone — that is precisely why such a matrix has no inverse. When the determinant is non-zero, the two-by-two inverse has a closed form worth memorising: swap the leading diagonal, negate the other two entries, divide by the determinant.

[a c b d]^{- 1} = \frac{1}{a d - b c} [d - c - b a]

Beyond two-by-two, use Gauss-Jordan elimination instead. Write the matrix beside an identity of the same size and apply row operations until the left half becomes the identity; whatever the right half has become is the inverse. Only three operations are permitted: swap two rows, multiply a row by a non-zero constant, and add a multiple of one row to another.

This machinery exists mostly to solve systems. Writing a linear system as $A x = b$ turns solving into one matrix operation.

A x = b ⟹ x = A^{- 1} b

In practice, row reducing the augmented matrix is faster than computing an inverse and then multiplying, and it keeps working when the system has no unique solution. The two inversion routes are compared in inverting a matrix two ways, and the connection to hand methods is in substitution, elimination, or graphing.

Worked examples

Worked example

[2013] [1 - 2 45]

1
Check the shapes first. The inner dimensions agree, so the product exists and is 2 by 2.
$(2 \times 2) (2 \times 2) \to 2 \times 2$
2
First row of the answer: pair row 1 of the left matrix with each column of the right matrix in turn.
$(2) (1) + (1) (- 2) = 0, (2) (4) + (1) (5) = 13$
3
Second row: the same pairing with row 2. The zero kills the first term of each product.
$(0) (1) + (3) (- 2) = - 6, (0) (4) + (3) (5) = 15$
4
Assemble the entries in position. Row index comes from the left matrix, column index from the right.
$[0 - 6 1315]$
5
Multiplying in the other order gives a different matrix entirely, which is the standard demonstration that matrix multiplication does not commute.
$[1 - 2 45] [2013] = [2 - 4 1313]$

Answer

[0 - 6 1315]

Worked example

A = [2513]

1
The determinant is 2·3 − 1·5 = 1, which is non-zero, so an inverse exists. Augment the matrix with the 2 by 2 identity.
$[25131001]$
2
Make the leading entry of row 1 into a 1 by halving the whole row, right-hand side included.
$R_{1} \to \frac{1}{2} R_{1} : [15 0.5 3 0.5 0 01]$
3
Clear the entry below it by subtracting 5 times row 1 from row 2.
$R_{2} \to R_{2} - 5 R_{1} : [10 0.5 0.5 0.5 - 2.5 01]$
4
Scale row 2 so its leading entry is 1. Doubling is the only operation needed here.
$R_{2} \to 2 R_{2} : [10 0.5 1 0.5 - 5 02]$
5
Clear the entry above the second pivot. The left half is now the identity, so the right half is the inverse.
$R_{1} \to R_{1} - \frac{1}{2} R_{2} : [1001 3 - 5 - 1 2]$
6
Confirm by multiplying. The product is the identity, which is the definition being satisfied.
$[2513] [3 - 5 - 1 2] = [1001]$

Answer

A^{- 1} = [3 - 5 - 1 2]

Try a matrix of your own

Change the four entries and watch the determinant and the inverse respond. Two experiments are worth running: make the second row a multiple of the first and see the determinant hit zero, then nudge one entry and watch the inverse reappear with enormous values — the matrix was nearly singular.

Determinant and inverse of a 2 by 2 matrix

Matrix A

Determinant

(2)(3) − (1)(5) = 1

Inverse

1/1 × swap diagonal, negate the other two

3-1-52

Common mistakes

Assuming the order does not matter

Matrix multiplication is not commutative. In the example above $A B$ and $B A$ differ in three of their four entries. When you multiply both sides of a matrix equation, you must do it on the same side of both.

Dividing by a matrix

There is no matrix division. To undo $A x = b$ you multiply by $A^{- 1}$ on the left of both sides, giving $x = A^{- 1} b$ and not $b A^{- 1}$ , which may not even be a legal product.

Dropping the alternating signs in a cofactor expansion

The middle term of an expansion along the top row is subtracted, not added. In the three-by-three example, using $+ 2 (0 \cdot 6 - 5 \cdot 1)$ instead of $- 2 (0 \cdot 6 - 5 \cdot 1)$ changes the determinant from 22 to 2.

Practice

Answers are checked here — nothing is sent anywhere.

1
$Find det [3412]$
2
$In [1324] [0110], find the row 2, column 1 entry$
3
$For which k is [k 6 34] singular?$

Frequently asked questions

What sizes can it handle?

Anything you would meet in a linear algebra course. Multiplication needs the inner dimensions to agree, and determinants and inverses need a square matrix; the working says which condition fails when one does.

Does it list the row operations?

Yes, one per line, written in the standard notation such as R2 → R2 − 5R1. That is the part markers award method credit for, and it is the part that is impossible to reconstruct from a final answer.

What happens if the matrix has no inverse?

The determinant comes out zero and the solution says the matrix is singular. Geometrically the transformation squashes space onto a line or a point, and that cannot be undone, which is why no inverse exists.

Can it find eigenvalues?

Ask for the characteristic equation, det(A − λI) = 0, and its roots. For a 2×2 matrix that polynomial is λ² − (trace)λ + determinant, so the two eigenvalues always sum to the trace and multiply to the determinant.

What does it cost?

The first solution is free with no account required. Free accounts get three a day, and Gauth Plus is $11.99 a month with a three-day free trial.

Keep going

Linear algebra solverMatrix operations, determinants, and systems with each row operation written out.Inverting a matrix two waysAdjugate against Gauss-Jordan, and how to tell when no inverse exists.Substitution, elimination, or graphing?Row reduction is elimination with the letters removed — see it both ways.System of equations solverSolve the same system without converting it to matrix form first.