By Anna Szczepanek, PhD
Last updated: May 14, 2021

Welcome to Omni's adjoint matrix calculator! Here you can quickly and easily determine the adjoint (a.k.a. adjugate) of a square matrix. Not sure what the adjoint of a matrix is? Need a quick reminder of how to find the adjoint of a matrix? In the article below we teach you all you need to know about the adjoint/adjugate matrix! We give the adjugate matrix formula and explain how to find the adjugate matrix of size 2x2 via this formula.

First of all, be aware that what we call the adjoint matrix here is sometimes called the adjugate matrix. You may also encounter the term classical adjoint matrix. This confusion stems from the fact that, in some contexts, the term adjoint can mean the conjugate transpose of a matrix, which is something entirely different from what we consider here. We will freely mix the terms adjoint and adjugate so that you could quickly get used to both of them.

How to find the adjoint of a matrix?

Suppose `A` is an `n × n` matrix with real or complex entries. To find the adjugate of `A`, follow these steps:

1. Delete the `i`-th row and the `j`-th column of `A`. What you get is a `(n - 1) × (n - 1)` submatrix of `A`.
2. Compute the determinant of this submatrix. What you get is called the `(i, j)`-minor of `A`.
3. Multiply the `(i, j)`-minor of `A` by the sign factor `(-1)i+j`. What you get is called the `(i, j)`-cofactor of `A`.
4. Repeat Steps 1, 2 and 3 for all `i,j = 1,...,n`.
5. The adjoint of `A` is the `n × n` matrix whose `(i, j)` entry is the `(j, i)` cofactor of `A`. Note, that the indices are flipped!

👉 The adjoint of matrix `A` is often denoted by `adj(A)`.

👉 If you are already familiar with the notion of cofactor matrix, then you may have realized that `adj(A)` is in fact the transpose of the cofactor matrix of `A`.

## Adjugate of a 2x2 matrix

Let's see how the adjugate matrix formula explained above works in the simplest case. Namely, we will use it to find the adjugate of a 2x2 matrix. Consider the following matrix:

 ⌈ a b ⌉ ⌊ c d ⌋
There are four coefficients, so we will repeat Steps 1, 2, and 3 from the previous section four times.
1. Let `i=1` and `j=1`.

Deleting the first row and the first column leaves us with a `1 x 1` matrix. Its single coefficient is equal to `d`. Hence, its determinant is also equal to `d`. The sign factor is given by `(-1)1+1 = 1`, so the `(1, 1)`-cofactor of our original matrix is equal to `d`.

2. Let `i=1` and `j=2`.

Deleting the first row and the second column leaves us with the `1 x 1` matrix containing `c`. The sign factor is given by `(-1)1+2 = -1`, so the `(1, 2)`-cofactor of our original matrix is equal to `-c`.

3. Let `i=2` and `j=1`.

Analogously, here we have the `1 x 1` matrix containing `b` and the sign factor is equal to `(-1)2+1 = -1`, so the `(2, 1)`-cofactor of our original matrix is equal to `-b`.

4. Let `i=2` and `j=2`.

Lastly, we deal with the `1 x 1` matrix containing `a`, the sign factor is equal to `(-1)2+2 = 1`, and so the `(2, 2)`-cofactor of the original matrix is equal to `a`.

Next, to find the adjugate of our matrix we put the `(i, j)`-cofactor into the `j`-th row and `i`-th column:

1. The `(1, 1)`-cofactor, which is equal to `d`, goes to the first row and first column:
 ⌈ d ⌉ ⌊ ⌋
1. The `(1, 2)`-cofactor, which is equal to `-c`, goes to the second row and first column:
 ⌈ d ⌉ ⌊ -c ⌋
1. The `(2, 1)`-cofactor, which is equal to `-b`, goes to the first row and second column:
 ⌈ d -b ⌉ ⌊ -c ⌋
1. The `(2, 2)`-cofactor, which is equal to `a`, goes to the second row and second column:
 ⌈ d -b ⌉ ⌊ -c a ⌋

That's it! We have derived a simple formula for the adjugate of a `2x2` matrix!

## How to use this adjugate matrix calculator?

Don't let the `2 x 2` case mislead you: calculating adjugate matrices by hand can be really time-consuming ⌛⌛ - especially if we have to deal with big matrices. Fortunately, our adjoint matrix calculator can do all this work for you! Here are the steps you should follow to use the adjugate matrix calculator efficiently:

1. Tell us the size of the matrix for which you need to find the adjugate.

2. Enter the coefficients of your matrix.

Tip: Our adjoint matrix calculator inputs the coefficients into the matrix as you enter them, so you can quickly verify if everything is OK.

3. The adjugate matrix appears at the bottom of the adjoint matrix calculator.

## Adjugate matrix formula for inverting a matrix

We now know how to find the adjugate matrix both by hand and with the help of the adjoint matrix calculator, but why do we even care about the adjugate matrix? For one reason, because it comes in very handy when we have to compute the inverse of a matrix.

Namely, when we multiply a matrix by its adjugate, we obtain the diagonal matrix whose diagonal entries are equal to the determinant of our matrix:

`A * adj(A) = det(A) * I,`

where `I` is the identity matrix of the same size as `A`. We can now calculate the inverse as:

`A-1 = (1 / det(A)) * adj(A).`

That is, to find the inverse of a matrix, you need to multiply its adjugate by the reciprocal of its determinant.

## FAQ

### How to calculate the adjugate matrix?

1. Find the cofactor of each entry.
2. Gather these cofactors in a matrix.
3. Transpose this matrix.
4. Enjoy the newly-found adjugate matrix.

### How to find the adjugate of a 2×2 matrix?

2. Swap the diagonal elements.
3. Change the signs of the anti-diagonal elements, i.e., the upper-right and the bottom-left element.
4. Congratulate yourself on finding the adjugate!

### How do I calculate the adjugate of a matrix transpose?

The adjugate of a matrix transpose, `AT`, is simply the transpose of the adjugate of `A`: `adj(AT) = adj(A)T`

### How do I calculate the adjugate of a product of matrices?

The adjugate of a product of two matrices `A` and `B` is equal to the product of their respective adjugates: `adj(AB) = adj(A)adj(B)`

### How do I calculate the adjugate of a matrix power?

The adjugate of a matrix power`Ak` is the `k`-th power of the adjugate of `A`: `adj(Ak) = adj(A)k`

### How to calculate the adjoint of a matrix?

Be aware there is some ambiguity in nomenclature. The adjoint of a matrix may refer to either its adjugate, i.e., the transpose of the cofactor matrix, or the conjugate transpose.

Anna Szczepanek, PhD
Matrix size
2 x 2
a₁a₂
b₁b₂
Enter the coefficients in the fields below.
Blanks are interpreted as zeros. Precision: 6 decimal places.
First row
a₁
a₂
Second row
b₁
b₂
00
00
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