Adjoint of a Matrix
The adjoint of a matrix is one of the easiest methods used to calculate the inverse of a matrix. Adjugate matrix is another term used to refer to the adjoint matrix in linear algebra. An adjugate matrix is especially useful in applications where an inverse matrix cannot be used directly.
The adjoint of a matrix is obtained by taking the transpose of the cofactor elements of the given matrix. In this article, let's learn about the adjoint of a matrix, its definition, properties with solved examples.
What Is an Adjoint of a Matrix?
The adjoint of a matrix B is the transpose of the cofactor matrix of B. The adjoint of a square matrix B is denoted by adj B. Let B = [\(b_{ij}\)] be a square matrix of order n. The three important steps involved in finding the adjoint of a matrix are:
 Find the minor matrix M of all the elements of the matrix B
 Find the cofactor matrix C of all the minor elements of the matrix M
 Find the adj B by taking the transpose of the cofactor matrix C
Adjoint of a square matrix
The adjoint adj(B) of a square matrix B of order n x n, can be defined as the transpose of the cofactor matrix. Consider the square matrix B with the elements \(b_{11}, b_{12}, b_{21}, b_{22}\), and its cofactor elements are \(B_{11}, B_{12}, B_{21}, B_{22}\) respectively.
Let's see an example of an adjoint of a 2×2 matrix.
Adjoint of a 2×2 Matrix
Consider the 2×2 matrix B:
\(B=\left[\begin{array}{ll}
3 & 6 \\
4 & 8
\end{array}\right]\)
The minor matrix M can be shown as:
\(M=\left\begin{array}{ll}
8 & 4 \\
6 & 3
\end{array}\right\)
The cofactor matrix C can be shown as:
\(C=\left[\begin{array}{ll}
8 & 4 \\
6 & 3
\end{array}\right]\)
The transpose C^{T} of the cofactor matrix can be shown as:
adj(B) = C^{T} = \(\left[\begin{array}{ll}
8 & 6 \\
4 & 3
\end{array}\right]\)
Let's see an example of an adjoint of a 3×3 matrix.
Adjoint of a 3×3 Matrix
Consider a 3×3 matrix:
\(B = \left[\begin{array}{ccc}
2 & 1 & 3 \\
0 & 5 & 2 \\
1 & 1 & 2
\end{array}\right] \)
The minor matrix M can be shown as:
\(M = \left\begin{array}{ccc}
8 & 2 & 5 \\
5 & 7 & 1 \\
17 & 4 & 10
\end{array}\right \)
The cofactor matrix C can be shown as:
\(C = \left[\begin{array}{ccc}
8 & 2 & 5 \\
5 & 7 & 1 \\
17 & 4 & 10
\end{array}\right] \)
The transpose T of the cofactor matrix can be shown as:
adj(B) = C^{T} = \(\left[\begin{array}{ccc}
8 & 5 & 17 \\
2 & 7 & 4 \\
5 & 1 & 10
\end{array}\right] \)
An adjoint of a matrix can be found only if we know the minor, cofactor, and the transpose of the cofactor matrix. Let's goahead to learn more about each of these in the below section.
Minor, Cofactor and Transpose of a Matrix
A matrix is a rectangular array that contains numbers or functions, arranged in rows and columns. These numbers are referred to as elements or entries of a matrix. It is essential to know about minors, transposes, and cofactors before learning about the adjoint of a matrix.
Minor of a Matrix
In a square matrix B, each element has its own minor. The minor is a value that is obtained from the determinant of a square matrix after deleting out a row and a column corresponding to that particular element of a matrix.
 Given a square matrix B, by minor of an element [\(b_{ij}\)], we refer to the value of the determinant obtained by deleting the i^{th} row and j^{th} column of the B matrix. It is denoted by \(M_{ij}\).
 To find the minor of any square matrix, we need to erase out a row and a column one by one at a time, and then calculate their determinant, until all the minors are calculated.
The following steps are to be followed to calculate the minor from any square matrix:
 Step 1: Hide the i^{th} row and j^{th} column one by one from the given matrix, here i refers to m and j refers to n, that is the total number of rows and columns in the matrix
 Step 2: Calculate the value of the determinant of the matrix made after hiding the row and the column obtained from Step 1.
Minor of a 2×2 Matrix
Description  Example 
Let us consider the 2×2 matrix B as: \(B=\left[\begin{array}{ll} 
Let us consider the 2×2 matrix C as: \(C=\left[\begin{array}{ll} 
Now, we need to hide the first row and the first column to find the minors of the matrix B \(\begin{array}{l} We have used (*) to denote the canceled row and column 
Now, we need to hide the first row and the first column to find the minors of the matrix B \(\begin{array}{l} We have used (*) to denote the canceled row and column 
The minor matrix M can be shown as:
\(M=\left[\begin{array}{ll}
8 & 4 \\
6 & 3
\end{array}\right]\)
Minor of a 3×3 Matrix
Description  Example 
Consider a 3×3 matrix: \(B = \left[\begin{array}{ccc} 
Consider a 3×3 matrix: \(B = \left[\begin{array}{ccc} 
Now, we need to hide the first row and the first column to find the minors of the matrix B \(\begin{array}{l}

Now, we need to hide the first row and the first column to find the minors of the matrix B \(\begin{array}{l} 
Cofactor of a Matrix
 A cofactor is a number we obtain by removing the column and row of a particular element in a matrix.
 The resultant is just a numerical grid that is in the form of a rectangle or a square.
 A positive (+) or negative () sign is used in front of the cofactor depending on whether the element in the matrix is in a positive (+) or () position.
Consider a square matrix B. By cofactor \(C_{ij}\) of an element \(b_{ij}\) of B, we mean minor of \(b_{ij}\) with a positive or negative sign depending on i and j.
Cofactor of a 2×2 Matrix
For a 2×2 matrix C, the negative sign is to be given to the minor elements \(b_{12}\) and \(b_{21}\) as:
\(C=\left[\begin{array}{ll}
+ &  \\
 & +
\end{array}\right]\)
Description  Example 
For a 2×2 matrix, the negative sign is to be given to the minor elements \(b_{12}\) and \(b_{21}\) as shown below: \(B=\left[\begin{array}{ll} 
Let us consider the 2×2 matrix B as: \(B=\left[\begin{array}{ll} 
The minor of \(M_{11}\), \(b_{11}\) is \(b_{22}\), and cofactor \(C_{11}\), \(b_{11}\) is \(b_{22}\) (sign unchanged) The minor of \(M_{12}\), \(b_{12}\) is \(b_{21}\) and cofactor \(C_{12}\), \(b_{12}\) is (\(b_{21}\)) (sign changed) The minor of \(M_{21}\), \(b_{21}\) is \(b_{12}\) and cofactor \(C_{21}\), \(b_{21}\) is (\(b_{12}\)) (sign changed) The minor of \(M_{22}\), \(b_{22}\) is \(b_{11}\) and cofactor \(C_{22}\), \(b_{22}\) is \(b_{11}\) (sign unchanged) 
The minor of \(M_{11}\), 3 is 2 and cofactor \(C_{11}\), 3 is 2 (sign unchanged) The minor of \(M_{12}\), 1 is 2 and cofactor \(C_{12}\), 1 is (2) (sign changed) The minor of \(M_{21}\), 2 is 1 and cofactor \(C_{21}\), 2 is (1) (sign changed) The minor of \(M_{22}\), 2 is 3 and cofactor \(C_{22}\), 2 is 3 (sign unchanged) 
Hence, the cofactor matrix \(C=\left[\begin{array}{ll}
2 & 2 \\
1 & 3
\end{array}\right]\)
Cofactor of a 3×3 Matrix
For a 3×3 matrix B, the negative sign is to be given to the minor elements as:
\(B = \left[\begin{array}{ccc}
+ &  & + \\
 & + &  \\
+ &  & +
\end{array}\right] \)
Consider the below example:
Consider the matrix B \(B = \left[\begin{array}{ccc} 
The minor matrix M can be shown as: \(M = \left\begin{array}{ccc} 
The cofactor matrix C can be shown as: \(C = \left[\begin{array}{ccc} 
Transpose of a Matrix
The matrix that is obtained from a given matrix B after interchanging its rows and columns is called the transpose of the matrix B.Transpose of B is denoted by B’ or B^{T}. If B is of order m*n, then B’ is of the order n*m. The transposed form of the transpose of B is the matrix B itself i.e. (B’)’= B.
Consider the matrix \(B = \left[\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array}\right] \)
Then, the transpose B^{T} of the matrix B, will be:
Consider the matrix B^{T} = \(\left[\begin{array}{ccc}
b_{11} & b_{21} & b_{31} \\
b_{12} & b_{22} & b_{32} \\
b_{13} & b_{23} & b_{33}
\end{array}\right] \)
Properties of Adjoint of Matrix
For any square matrix of order n x n, the following properties of the adjoint of a matrix can be applied. Let us consider a matrix B here:
 If 0 is a null matrix and I is an identity matrix then, adj (0) = 0 and adj (I) =I
 adj(B^{T}) = adj(B)^{T}, here B^{T} is a transpose of a matrix B
 The adjoint of a matrix B can be defined as the product of B with its adjoint yielding a diagonal matrix whose diagonal entries are the determinant det(B). B adj(B) = adj(B) B = det(B) I, where I is an identity matrix.
 Suppose C is another square matrix then, adj(BC) = adj(C) adj(B)
 For any nonnegative integer k, adj(B^{k}) = adj(B)^{k}
Adjoint of a Diagonal Matrix
In linear algebra, a matrix is considered to be diagonal if all the entries outside the main diagonal are zero. The adjoint of any diagonal matrix will yeild a diagonal matrix again. Let us see the examples of 2×2 diagonal matrix to understand this better:
Consider the 2×2 matrix B:
\(B=\left[\begin{array}{ll}
3 & 0 \\
0 & 8
\end{array}\right]\)
The minor matrix M can be shown as:
\(M=\left\begin{array}{ll}
8 & 0 \\
0 & 3
\end{array}\right\)
The cofactor matrix C can be shown as:
\(C=\left[\begin{array}{ll}
8 & 0 \\
0 & 3
\end{array}\right]\)
The transpose C^{T} of the cofactor matrix can be shown as:
adj(B) = C^{T} = \(\left[\begin{array}{ll}
8 & 0 \\
0 & 3
\end{array}\right]\)
In this example, we can see that adjoint of the diagonal matrix B yielded a diagonal matrix again.
Relation Between Adjoint and Inverse of a Matrix
Consider a matrix B, and another matrix C such that, B × C = C × B = I, then C is called as the inverse of B. When a number is multiplied by its reciprocal, then we will get 1. Same way, when we multiply a matrix by its inverse, we will get an Identity matrix. The inverse of a matrix is usually used to find the solution to a system of linear equations. Determinants and adjoints are used to find the inverse of a square matrix. The inverse of B is denoted by B^{1}. The relationship between the adjoint adj(B) and the inverse of a matrix B^{1} is represented as B^{1} = (1/B) × adj(B). Here are the steps to be followed to calculate the inverse of a matrix:
 Find the determinant of B. If B = 0, then the inverse does not exist. Only if B \(\neq\) 0, the inverse exists.
 Find the minor matrix M of all the elements of the matrix B
 Find the cofactor matrix C of all the minor elements of the matrix M
 Find the adj B by taking the transpose of the cofactor matrix C
 Then, find the inverse of the matrix B as B^{1} = (1/B) × adj(B)
 Check if the inverse is correct by verifying it as B× B^{1} = I, where I is an identity matrix.
The description and an example of the inverse of a 2×2 matrix are given in the below table:
Description  Example 
Suppose B is a matrix of order 2× 2 and its determinant is not equal to zero: \(B=\left[\begin{array}{ll} then, the inverse of the matrix B will be \(B^{1}=\left[\begin{array}{ll} 
Let's find the inverse of this matrix: \(B=\left[\begin{array}{ll} then, the inverse of the matrix B will be \(B^{1}=\left[\begin{array}{ll} 
The first step is to check if the determinant of the matrix is not equal to zero. So, B = (ad  bc) 
The first step is to check if the determinant of the matrix is not equal to zero. B = ((2)(2))  ((1)(3)) = 4 + 3 = 1 \(\neq\) 0 
Then, find the minor and cofactor of B, then take the transpose of the cofactor of B to obtain the adjoint

Then, find the minor of B as: \(M=\left\begin{array}{ll} and the cofactor of B as: \(C=\left[\begin{array}{ll} then take the transpose of the cofactor of B to obtain the adjoint adj(B) = C^{T} = \(\left[\begin{array}{ll} 
Then, finally apply this formula: B^{1} = \(\frac{1}{a db c}\left[\begin{array}{cc} 
B^{1} = \(\frac{1}{1}\left[\begin{array}{cc} =\(\left[\begin{array}{ll} 
Relation Between Determinant of a Matrix and its Adjoint
In every square matrix B = [\(b_{ij}\)] of order n × n, a real or a complex number called as determinant can be associated to the square matrix B, where \(b_{ij}\) = (i,j)^{th} element of B. It is denoted by B or det B. The relationship between a determinant of a matrix B and its adjoint adj(B) can be shown as B × adj(B) = adj(B) × B = B × I. Here, B is a square matrix and I is an identity matrix. The description and an example of the determinant of a 2×2.
Suppose B is a matrix of order 2×2 matrix such as:
\(B=\left[\begin{array}{ll}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array}\right]\)
then, the determinant of the matrix B will be
\(\mathrm{B}=\left\begin{array}{ll}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array}\right\)
B = (\(b_{11}\) × \(b_{22}\))  (\(b_{21}\) × \(b_{12}\))
Let's find the determinant of this matrix:
\(B=\left[\begin{array}{ll}
4 & 1 \\
3 & 2
\end{array}\right]\)
then, the determinant of this matrix is:
\(\mathrm{B}=\left\begin{array}{ll}
4 & 1 \\
3 & 2
\end{array}\right\)
B = (4 × 2)  (3 × 1)
= 8  3
= 5
Related Articles on Adjoint of MatrixCheck out the following pages related to the adjoint of a matrix.
 Matrix Calculator
 Matrix formula
 How to Solve Matrices
 Diagonal Matrix Calculator
 Transpose Matrix Calculator
Important Notes on Adjoint of Matrix
Here is a list of a few points that should be remembered while studying the adjoint of a matrix:
 The minor is a value that is obtained from the determinant of a square matrix after deleting out a row and a column corresponding to that particular element of a matrix.
 In a transpose matrix, the rows of the original matrix become the columns of the transposed matrix, and the columns of the original matrix become the rows of a transposed matrix.
 The adjoint of a matrix is formed by taking the transpose of the matrix of the cofactors.
Examples on Adjoint of a Matrix

Example 1: Find the adjoint of the matrix
\(A=\left[\begin{array}{ll}
4 & 5\\
2 & 1
\end{array}\right]\)Solution: The solution can be found as follows:
Consider the given 2×2 matrix A:
\(A=\left[\begin{array}{ll}
4 & 5\\
2 & 1
\end{array}\right]\)The minor matrix M can be shown as:
\(M=\left\begin{array}{ll}
1 & 2 \\
5 & 4
\end{array}\right\)The cofactor matrix C can be shown as:
\(C=\left[\begin{array}{ll}
1 & 2 \\
5 & 4
\end{array}\right]\)The transpose C^{T} of the cofactor matrix can be shown as:
adj(A) = C^{T} = \(\left[\begin{array}{ll}
1 & 5 \\
2 & 4
\end{array}\right]\)Answer: Therefore Adj (A) = \(\left[\begin{array}{ll}
1 & 5 \\
2 & 4
\end{array}\right]\) 
Example 2: Find the inverse of the matrix B by taking the adjoint of the matrix B.
\(B=\left[\begin{array}{ll}
2 & 4\\
3 & 5
\end{array}\right]\)Solution:
The given matrix is a 2 x 2 matrix, and hence it is easy to find the inverse of this square matrix. First we need to find the determinant of this matrix, and then find the adjoint of this matrix, to find the inverse of the matrix.
\(B=\left[\begin{array}{ll}
2 & 4\\
3 & 5
\end{array}\right]\)det B = B = 2 x 5  4 x 3 = 10  12 = 2
\(adj(B)=\left[\begin{array}{ll}
5 & 4\\
3 & 2
\end{array}\right]\)B^{1} = \(\dfrac{1}{B}.Adj(B)\)
B^{1} = \(\dfrac{1}{2}.\left[\begin{array}{ll}
5 & 4\\
3 & 2
\end{array}\right]\)Answer: Therefore B^{1} = \(\dfrac{1}{2}.\left[\begin{array}{ll}
5 & 4\\
3 & 2
\end{array}\right]\)
FAQs on Adjoint of a Matrix
What Is Adjoint of a Matrix?
The adjoint of a matrix B is the transpose of the cofactor matrix of B. The adjoint of a square matrix B is denoted by adj B. Let B = [\(b_{ij}\)] be a square matrix of order n. Consider the example of the matrix B:
\(B=\left[\begin{array}{ll}
3 & 6 \\
4 & 8
\end{array}\right]\)
The adjoint for a given matrix B is:
adj(B) = C^{T} = \(\left[\begin{array}{ll}
8 & 6 \\
4 & 3
\end{array}\right]\)
How Do You Find the Adjoint of a Matrix?
There are 3 steps to be followed in order to find the adjoint of a matrix:
 Find the minor matrix M of all the elements of the original matrix
 Find the cofactor matrix C of all the minor elements of the matrix M
 Find the adjoint by taking the transpose of the cofactor matrix C. The resultant matrix will be the adjoint of the original matrix.
What Are the Different Properties of the Adjoint of Matrix?
For any square matrix of order n x n, the following properties of the adjoint of a matrix can be applied. Let us consider a matrix B here:
 If 0 is a null matrix and I is an identity matrix then, adj (0) = 0 and adj (I) =I
 adj(B^{T}) = adj(B)^{T}, here B^{T} is a transpose of a matrix B
 The adjoint of a matrix B can be defined as the product of B with its adjoint yielding a diagonal matrix whose diagonal entries are the determinant det(B). B adj(B) = adj(B) B = det(B) I, where I is an identity matrix.
 Suppose C is another square matrix then, adj(BC) = adj(C) adj(B)
 For any nonnegatie integer k, adj(B^{k}) = adj(B)^{k}
What Is the Adjoint of a Square Matrix?
The adjoint adj(B) of a square matrix B of order n*n, can be defined as the transpose of the cofactor matrix. Consider the square matrix B with these elements:
Suppose B = \(\left[\begin{array}{ll}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array}\right]\)
Then, the adj(B) is shown as:
adj (B) = \(\left[\begin{array}{ll}
B_{11} & B_{21} \\
B_{12} & B_{22}
\end{array}\right]\)
Why Do We Use the Adjoint of a Matrix?
The adjoint of a matrix is one of the easiest methods used to calculate the inverse of a matrix. Adjugate matrix is another term used to refer to the adjoint matrix in linear algebra. An adjugate matrix is especially useful in applications where an inverse matrix cannot be used directly.
How Do You Find the Adjoint of a 2 × 2 Matrix?
For any 2 × 2 matrices, these are the steps involved in finding the adjoint of the matrix:
 Find the minor matrix M of all the elements of the 2 × 2 matrix
 Find the cofactor matrix C of all the minor elements of the matrix M
 Find the adj by taking the transpose of the cofactor matrix C. adj(B) = C^{T}
How To Find Inverse of a Matrix by Adjoint Method?
Here are the steps to be followed to calculate the inverse of a matrix B by using the adjoint method:
 Find the determinant of B. If B = 0, then the inverse does not exist. Only if B \(\neq\) 0, the inverse exists.
 Find the minor matrix M of all the elements of the matrix B
 Find the cofactor matrix C of all the minor elements of the matrix M
 Find the adj B by taking the transpose of the cofactor matrix C
 Then, find the inverse of the matrix B as B^{1} = (1/B) × adj(B)
 Check if the inverse is correct by verifying it as B× B^{1} = I, where I is an identity matrix.