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 co-factor elements of the given matrix. In this article, let's learn about the adjoint of a matrix, its definition, properties with solved examples.

1.	What is an Adjoint of a Matrix?
2.	Minor, Cofactor and Transpose of a Matrix
3.	Adjoint of a 2×2 Matrix
4.	Adjoint of a 3×3 Matrix
5.	Properties of Adjoint of Matrix
6.	Finding Inverse Using Adjoint of a Matrix
7.	FAQs on Adjoint of 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 matrix B.
• Find the cofactor matrix C of all the minor elements of matrix M.
• Find the adj B by taking the transpose of the cofactor matrix C.

Adjoint of a Matrix Formula

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 2x2 matrix B with the elements \(b_{11}, b_{12}, b_{21}, b_{22}\), and their cofactor elements are \(B_{11}, B_{12}, B_{21}, B_{22}\) respectively. Then the adjoint of matrix formula is as follows:

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

Minor of 2x2 Matrix	Example
Let us consider the 2×2 matrix B as: \(B=\left[\begin{array}{ll} p & q \\ \\ r & s \end{array}\right]\)	Let us consider the 2×2 matrix A as: \(A=\left[\begin{array}{ll} 3 & 6 \\ \\ -4 & 8 \end{array}\right]\)
Now, we need to hide the first row and the first column to find the minors of the matrix B \(\begin{array}{l} M_{11}=\left|\begin{array}{ll} * & *\\ * & s \end{array}\right|={s} \\ M_{12}=\left|\begin{array}{ll} * & * \\ r & * \end{array}\right|={r} \\ M_{21}=\left|\begin{array}{ll} * & q \\ * & * \end{array}\right|={q} \\ M_{22}=\left|\begin{array}{ll} p & * \\ * & * \end{array}\right|={p} \end{array}\) 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 \(\begin{array}{l} M_{11} = \text{Minor of }3 =\left|\begin{array}{ll} * & * \\ * & 8 \end{array}\right| \\ = {8} \\ M_{12} = \text{Minor of 6} =\left|\begin{array}{ll} * & * \\ -4 & * \end{array}\right|\\={-4} \\ M_{21} = \text{Minor of }-4 =\left|\begin{array}{ll} * & 6 \\ * & * \end{array}\right| \\ ={6} \\ M_{22} = \text{Minor of 8} =\left|\begin{array}{ll} 3 & * \\ * & * \end{array}\right|\\={3} \end{array}\) 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

Minor of 3x3 Matrix	Example
Consider a 3×3 matrix: \(B = \left[\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \)	Consider a 3×3 matrix: \(B = \left[\begin{array}{ccc} 2 & -1 & 3 \\ 0 & 5 & 2 \\ 1 & -1 & -2 \end{array}\right] \)
Now, we need to hide the first row and the first column to find the minors of the matrix B \(\begin{array}{l} M_{11}=\left|\begin{array}{ll} e & f \\ h & i \end{array}\right|={ei}-{hf} \\ M_{12}=\left|\begin{array}{ll} d & f \\ g & i \end{array}\right|={di}-{fg} \\ M_{13}=\left|\begin{array}{ll} d & e \\ g & h \end{array}\right|={dh}-{eg} \\ M_{21}=\left|\begin{array}{ll} b & c \\ h & i \end{array}\right|={bi}-{ch} \\ M_{22}=\left|\begin{array}{ll} a & c \\ g & i \end{array}\right|={ai}-{cg} \\ M_{23}=\left|\begin{array}{ll} a & b \\ g & h \end{array}\right|={ah}-{bg} \\ M_{31}=\left|\begin{array}{ll} b & c \\ e & f \end{array}\right|={bf}-{ce} \\ M_{32}=\left|\begin{array}{ll} a & c \\ d & f \end{array}\right|={af}-{cd} \\ M_{33}=\left|\begin{array}{ll} a & b \\ d & e \end{array}\right|={ae}-{bd} \end{array}\)	Now, we need to hide the first row and the first column to find the minors of the matrix B \(\begin{array}{l} M_{11}=\left|\begin{array}{ll} 5 & 2 \\ -1 & -2 \end{array}\right|\\={((5)(-2))}-{((-1)(2))} \\= {(-10)}-{(-2)} = {-8} \\ M_{12}=\left|\begin{array}{ll} 0 & 2 \\ 1 & -2 \end{array}\right|\\={((0)(-2))}-{((1)(2))} \\= {(0)}-{(2)} = {-2} \\ M_{13}=\left|\begin{array}{ll} 0 & 5 \\ 1 & -1 \end{array}\right|\\={((0)(-1))}-{((1)(5))} \\= {(0)}-{(5)} = {-5} \\ M_{21}=\left|\begin{array}{ll} -1 & 3 \\ -1 & -2 \end{array}\right|\\={((-1)(-2))}-{((-1)(3))} \\= {(2)}-{(-3)} = {5} \\ M_{22}=\left|\begin{array}{ll} 2 & 3 \\ 1 & -2 \end{array}\right|\\={((2)(-2))}-{((1)(3))} \\= {(-4)}-{(3)} = {-7} \\ M_{23}=\left|\begin{array}{ll} 2 & -1 \\ 1 & -1 \end{array}\right|\\={((2)(-1))}-{((1)(-1))} \\= {(-2)}-{(-1)} = {-1} \\ M_{31}=\left|\begin{array}{ll} -1 & 3 \\ 5 & 2 \end{array}\right|\\={((-1)(2))}-{((5)(3))} \\= {(-2)}-{(15)} = {-17} \\ M_{32}=\left|\begin{array}{ll} 2 & 3 \\ 0 & 2 \end{array}\right|\\={((2)(2))}-{((0)(3))} \\= {(4)}-{(0)} = {4} \\ M_{33}=\left|\begin{array}{ll} 2 & -1 \\ 0 & 5 \end{array}\right|\\={((2)(5))}-{((0)(-1))} \\= {(10)}-{(0)} = {10} \end{array}\)

Hence, the minor matrix of B is, \(M = \left[\begin{array}{ccc}
-8 & -2 & -5 \\
5 & -7 & -1 \\
-17 & 4 & 10
\end{array}\right]\).

Cofactor of a Matrix

The cofactor of an element a_ij is obtained by multiplying its corresponding minor by (-1)^i+j. i.e., the minor is multiplied by a positive (+) or negative (-) sign 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

To find the cofactors of 2x2 matrix, the corresponding minors should be multiplied the signs below according to their position.

\(C =\left[\begin{array}{ll}
+ & - \\
- & +
\end{array}\right] \)

We have seen that the minor matrix of A = \(\left[\begin{array}{ll}
3 & 6 \\ \\
-4 & 8
\end{array}\right]\) is \(M=\left[\begin{array}{ll}
8 & -4 \\ \\
6 & 3
\end{array}\right]\). Multiplying the cofactors by the above signs, we get the cofactor matrix to be, C = \(\left[\begin{array}{ll}
8 & 4 \\
-6 & 3
\end{array}\right]\)

Cofactor of a 3×3 Matrix

To find the cofactors of 2x2 matrix, the corresponding minors should be multiplied the signs below according to their position.

\(C=\left[\begin{array}{lll}
+ & - & + \\
- & + & - \\
+ & - & +
\end{array}\right]\)

We have seen that the minor matrix of \(B = \left[\begin{array}{ccc}
2 & -1 & 3 \\
0 & 5 & 2 \\
1 & -1 & -2
\end{array}\right] \) is \(M = \left[\begin{array}{ccc}
-8 & -2 & -5 \\
5 & -7 & -1 \\
-17 & 4 & 10
\end{array}\right]\).

Multiplying the cofactors by the above signs, we get the cofactor matrix to be, C = \( \left[\begin{array}{ccc}
-8 & 2 & -5 \\
-5 & -7 & 1 \\
-17 & -4 & 10
\end{array}\right]\).

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 = \(\left[\begin{array}{ccc}
b_{11} & b_{21} & b_{31} \\
b_{12} & b_{22} & b_{32} \\
b_{13} & b_{23} & b_{33}
\end{array}\right] \)

Let's see an example of an adjoint of a 2×2 matrix. Consider the 2×2 matrix A (that we had considered earlier):

\(A=\left[\begin{array}{ll}
3 & 6 \\ \\
-4 & 8
\end{array}\right]\)

We have already found its cofactor matrix to be C = \(\left[\begin{array}{ll}
8 & 4 \\
-6 & 3
\end{array}\right]\).

By transposing this, we get the adjoint of 2x2 matrix A.

Then adj A = 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. Consider a 3×3 matrix B (which we considered earlier):

\(B = \left[\begin{array}{ccc}
2 & -1 & 3 \\
0 & 5 & 2 \\
1 & -1 & -2
\end{array}\right] \)

We already found its cofactor matrix to be C = \( \left[\begin{array}{ccc}
-8 & 2 & -5 \\
-5 & -7 & 1 \\
-17 & -4 & 10
\end{array}\right]\).

By taking its transpose, we get the adjoint of 3x3 matrix B. Then

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 go-ahead to learn more about each of these in the below section.

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 non-negative integer k, adj(B^k) = adj(B)^k.
• The adjoint of a diagonal matrix is a diagonal matrix again.

The inverse of a matrix A, which is represented as A^-1, is found using the adjoint of matrix. Its formula is A^-1 = (1/|A|) × adj(A). Here,

• |A| = the determinant of A
• adj(A) = adjoint of A.

We already found that for \(A=\left[\begin{array}{ll}
3 & 6 \\ \\
-4 & 8
\end{array}\right]\), the adjoint matrix is, adj A = C^T = \(\left[\begin{array}{ll}
8 & -6 \\ \\
4 & 3
\end{array}\right]\).

Its determinant is, |A| = 3(8) - 6(-4) = 24 + 24 = 48.

So the inverse of the matrix A is,

A^-1 = (1/|A|) × adj(A) = 1/48 \(\left[\begin{array}{ll}
8 & -6 \\ \\
4 & 3
\end{array}\right]\) = \(\left[\begin{array}{ll}
1/6 & -1/8 \\ \\
1/12 & 1/16
\end{array}\right]\).

☛ Related Topics:

• Matrix Calculator
• Matrix formula
• 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.

 

Define Adjoint of a Matrix.

The adjoint of a matrix A is equal to the transpose of the cofactor matrix of A. The adjoint of a square matrix B is denoted by adj B. 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) = \(\left[\begin{array}{ll}
8 & -6 \\
4 & 3
\end{array}\right]\).

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(A^T) = adj(A)^T
• A adj(A) = adj(A) A = det(A) I, where I is an identity matrix.
• adj(AB) = adj(B) adj(A)
• adj(A^k) = adj(A)^k, where 'k' is a non-negative integer.

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 Adjugate Matrix?

The adjoint of a matrix is used to calculate the inverse of a matrix. It is also known as the adjugate matrix. An adjugate matrix is useful in finding the inverse matrix.

How Do You Find the Adjoint of a 2 × 2 Matrix?

Here are the steps involved in finding the adjoint of a 2x2 matrix A:

• Find the minor matrix M by finding minors of all elements.
• Find the cofactor matrix C by multiplying elements of M by (-1)^{row number + column number}.
• Then the adjoint matrix is, adj(A) = C^T.

Alternatively, we can use the formula: adjoint of a matrix A = \(\left[\begin{array}{ll}
a & b \\ \\
c & d
\end{array}\right] \) is adj A = \(\left[\begin{array}{ll}
d &-b \\ \\
-c & a
\end{array}\right] \).

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| ≠ 0, the inverse exists.
• Find the minor matrix M.
• Find the cofactor matrix C.
• 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.