Adjoint of a matrix (adjugate matrix)

On this post we explain what the adjoint of a matrix is and how to find it. Also, you will see several examples of adjoint of matrices and, finally, all the properties of this type of matrix.

What is the adjoint of a matrix?

The definition of adjoint of a matrix is as follows:

The adjoint of a matrix, also known as adjugate matrix, is the transpose of its cofactor matrix.

The adjoint of a matrix is also called classical adjoint of a matrix or adjunct matrix. Moreover, the adjoint of a matrix is denoted by adj(A).

How to find the adjoint of a matrix

Once we’ve seen the meaning of the adjoint of a matrix, let’s see how to calculate it:

To find the adjoint of a matrix, first replace each element in the matrix by its cofactor and then transpose the matrix.

Remember that the formula to compute the i, j cofactor of a matrix is as follows:

C_{i,j}=(-1)^{i+j}\cdot M_{ij}

Where Mij is the i, j minor of the matrix, that is, the determinant that results from deleting the i-th row and the j-th column of the matrix.

Note that the adjoint of a matrix can only be found for square matrices.

Examples of adjoint of matrices

Having seen the theory of the adjoint of a matrix, here are some solved examples of the calculation of the adjoint of a matrix. We will first see the adjoint of a 2×2 dimension matrix, and then the adjoint of a 3×3 dimension matrix.

Example of a 2×2 matrix

Let A be the following square matrix of order 2:

\displaystyle A =\begin{pmatrix} 4 & -1 \\[1.1ex] 3 & 2  \end{pmatrix}

To compute the adjoint of matrix A, we first have to find the cofactor of each entry of the matrix. To do this, we have to apply the following formula:

C_{i,j}=(-1)^{i+j}\cdot M_{ij}

\text{Cofactor of 4} \ \longrightarrow \ \displaystyle C_{11}=(-1)^{1+1} \cdot \begin{vmatrix} 2 \end{vmatrix}=1\cdot2=2

\text{Cofactor of -1} \ \longrightarrow \ \displaystyle C_{12}=(-1)^{1+2} \cdot\begin{vmatrix} 3 \end{vmatrix}=-1\cdot3=-3

\text{Cofactor of 3} \ \longrightarrow \ \displaystyle C_{21}=(-1)^{2+1} \cdot\begin{vmatrix} -1 \end{vmatrix}=-1\cdot(-1)=1

\text{Cofactor of 2} \ \longrightarrow \ \displaystyle C_{22}=(-1)^{2+2} \cdot\begin{vmatrix} 4 \end{vmatrix}=1\cdot4=4

Now we replace each element of matrix A by its cofactor to find the cofactor matrix of A:

\begin{pmatrix}2& -3\\[1.1ex]1&4\end{pmatrix}

And finally, we simply have to transpose the cofactor matrix:

\displaystyle\text{Adj}(A)=\begin{pmatrix}2& 1\\[1.1ex]-3&4\end{pmatrix}

On the other hand, there is a formula to find the adjoint of a 2×2 matrix without doing any calculations:

\displaystyle A=\begin{pmatrix}a&b\\[1.1ex]c&d\end{pmatrix}\ \longrightarrow \ \text{Adj}(A)=\begin{pmatrix}d&-b\\[1.1ex]-c&a\end{pmatrix}

However, this formula is only valid for 2×2 matrices. You can verify the formula by calculating with it the example seen above.

Example of a 3×3 matrix

Let B be the following square matrix of order 3:

\displaystyle B=\begin{pmatrix} 1 & 3 & -1 \\[1.1ex] 2 & 4 & 0 \\[1.1ex] 5 & 0 & -2 \end{pmatrix}

To compute the adjoint of the 3×3 matrix we have to apply the same procedure. So, first we find the cofactor of each element of the matrix:

C_{i,j}=(-1)^{i+j}\cdot M_{ij}

\text{Cofactor of 1} \ \longrightarrow \ \displaystyle C_{11}=(-1)^{1+1}\cdot\begin{vmatrix} 4 & 0 \\[1.1ex] 0 & -2\end{vmatrix} = 1 \cdot (-8) =-8

\text{Cofactor of 3} \ \longrightarrow \ \displaystyle C_{12}= (-1)^{1+2}\cdot\begin{vmatrix}  2 & 0 \\[1.1ex] 5 & -2\end{vmatrix} = -1 \cdot (-4) = 4

\text{Cofactor of -1} \ \longrightarrow \ \displaystyle C_{13}= (-1)^{1+3}\cdot\begin{vmatrix} 2 & 4 \\[1.1ex] 5 & 0\end{vmatrix} = 1 \cdot (-20) =-20

\text{Cofactor of 2} \ \longrightarrow \ \displaystyle C_{21}= (-1)^{2+1}\cdot\begin{vmatrix} 3 & -1 \\[1.1ex] 0 & -2\end{vmatrix} = -1 \cdot (-6)=6

\text{Cofactor of 4} \ \longrightarrow \ \displaystyle C_{22}= (-1)^{2+2}\cdot\begin{vmatrix} 1 & -1 \\[1.1ex] 5 & -2\end{vmatrix} = 1 \cdot 3=3

\text{Cofactor of 0} \ \longrightarrow \ \displaystyle C_{23}=(-1)^{2+3}\cdot\begin{vmatrix} 1 & 3  \\[1.1ex] 5 & 0 \end{vmatrix} = -1 \cdot (-15)=15

\text{Cofactor of 5} \ \longrightarrow \ \displaystyle C_{31}= (-1)^{3+1}\cdot\begin{vmatrix} 3 & -1 \\[1.1ex] 4 & 0 \end{vmatrix} = 1 \cdot 4=4

\text{Cofactor of 0} \ \longrightarrow \ \displaystyle C_{32}= (-1)^{3+2}\cdot\begin{vmatrix} 1 & -1 \\[1.1ex] 2 & 0\end{vmatrix} = -1 \cdot 2 =-2

\text{Cofactor of -2} \ \longrightarrow \ \displaystyle C_{33}=(-1)^{3+3}\cdot\begin{vmatrix} 1 & 3 \\[1.1ex] 2 & 4 \end{vmatrix} = 1 \cdot (-2) =-2

Secondly, we replace each element of matrix B by its cofactor to determine the cofactor matrix of B:

\begin{pmatrix}-8&4&-20\\[1.1ex]6&3&15\\[1.1ex]4&-2&-2\end{pmatrix}

And finally, we transpose the cofactor matrix to find the adjoint of matrix B:

\displaystyle\text{Adj}(B)=\begin{pmatrix}-8&6&4\\[1.1ex]4&3&-2\\[1.1ex]-20&15&-2\end{pmatrix}

There is no formula to directly find the adjoint of a 3×3 matrix.

➤ Did you know that the adjugate matrix is used to calculate the inverse of a matrix? See how to find the matrix inverse.

Properties of the adjoint of a matrix

The adjoint of a matrix has the following characteristics:

  • The adjoint of the zero matrix (or null matrix) results in the zero matrix:

\displaystyle\text{adj}(0)=0

  • Likewise, the adjoint of the identity matrix of any order results in the identity matrix (of the same order).

\displaystyle\text{adj}(I)=I

  • Transposing a matrix first and then finding its adjoint is the same as first finding the adjoint of the matrix and then transposing the result.

\displaystyle\text{adj}(A^T)=\text{adj}(A)^T

  • The determinant of the adjoint of a matrix equals to the determinant of the matrix raised to n-1, where n is the order of the matrix.

\displaystyle\text{det}\bigl(\text{adj}(A)\bigr)=\bigl(\text{det}(A)\bigr)^{n-1}

  • If matrix A is invertible, then the adjoint of matrix A is equal to the product of the determinant of matrix A and the inverse of matrix A.

\displaystyle\text{adj}(A^T)=\text{det}(A)\cdot A^{-1}

  • If a matrix is invertible, then we can find the inverse of the adjoint of the matrix using the following formula:

\displaystyle\bigl(\text{adj}(A)\bigr)^{-1}=\cfrac{1}{\text{det}(A)}\cdot A

  • In addition, if a matrix is invertible, calculating the inverse of the matrix first and then its adjoint is the same as calculating its adjoint first and then inverting the matrix.

\displaystyle\text{adj}(A^{-1})=\text{adj}(A)^{-1}

  • The adjoint of a matrix multiplication equals to the product of the adjoint of each matrix but multiplied in different order:

\displaystyle\text{adj}(A\cdot B)=\text{adj}(B)\cdot \text{adj}(A)

  • The adjoint of a scalar multiplication is equal to the product of the scalar raised to n-1 and the adjoint of the matrix, where n is the order of the matrix.

\displaystyle\text{adj}(cA)=c^{n-1}\cdot \text{adj}(A)

  • Let A be a square matrix of order n, if the rank of matrix A is less than or equal to n-2, then the adjoint of matrix A results in 0.

\displaystyle rk(A)\leq n-2 \ \longrightarrow \ \text{adj}(A)=0

  • Let A be a square matrix of order n, if the rank of matrix A is n-1, then the rank of the adjoint of matrix A is 1.

\displaystyle rk(A)=n-1 \ \longrightarrow \ rk\bigl(\text{adj}(A)\bigr)=1

Leave a Comment

Your email address will not be published.

Scroll to Top