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The adjoint matrix of linear algebra is also referred to as the adjugate matrices. When an inverse matrix can't be used directly, an adjugate matrix is extremely helpful.
- By transposing the cofactor elements in the provided matrix, one can obtain the adjoint of a matrix.
- It is used in many facets of life and business, including sales, finance, and science.
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Key Terms: Cofactor, Minor, Matrix, Transpose, Identity, Null
Adjoint of a Matrix
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Aij is the cofactor of the element aij, and the adjoint of a square matrix A = [aij]nxn is defined as the matrix [aij]nxn transposed. In other words, the adjoint of the matrix is the transposition of a cofactor matrix of the square matrix. adj A stands for the adjoint of matrix A.
- Adjugate matrix or adjunct matrix are other names for this.
- To calculate the inverse matrix, one needs to locate the adjoint of a given matrix.
- Only square matrices allow for this to be done.
Calculating Adjoint of a Matrix
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These procedures must be taken in order to compute a matrix A's adjoint:
Adjoint of a 2 x 2 matrix
The cofactor of the 2x2 matrix can be found by interchanging a11 and a22 and changing the signs of a12 and a21. Later the transpose of the cofactor matrix is taken in order to get the adjoint matrix
Adjoint of a 2 x 2 matrix
Adjoint of a 3 x 3 matrix
Calculating the determinant of the matrix created by eliminating the ith row and jth column of A for each element aij in the matrix of minors.
- To get the matrix of cofactors, multiply each of the elements in the matrix of minors by (-1)i+j.
- To create the adjoint matrix, transpose the cofactor matrix.
Minor of a Matrix
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Each element in a square matrix B has a unique minor. After eliminating the row and column that correspond to that specific matrix element, the minor value can be calculated from the determinant of a square matrix. To determine the minor from any square matrix, use the procedures below:
- Step 1: In the supplied matrix, hide the ith row and jth column one at a time. Here, i stands for m and j for n, the total number of rows and columns in the matrix.
- Step 2: After hiding the row and column from Step 1's matrix, determine the value of the determinant that remains.
Minor of a Matrix
Cofactor of a Matrix
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By dividing the minor of an element by (-1)i+j, one can get the cofactor of an element aij. i.e., depending on the fact that the element in the matrix is in a positive (+) or (-) position, the minor is multiplied by a positive (+) or negative (-) sign. We refer to the minor of an element bij of B with a positive or negative sign depending on i and j as the cofactor Cij.
Transpose of a Matrix
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A matrix that has been transposed is one that has had its rows and columns switched around, or vice versa, meaning that for the given matrix, the elements in the rows have been switched with the elements in the columns. The transpose of any given matrix A is abbreviated as At or AT.
Properties of Adjoint Matrix
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The properties of the Adjoint Matrix are as follows:
- The adjoint of a matrix can have the following properties for any of a square matrix of order n x n.
- Adj (0) = 0 and Adj (I) = I if I is an identity matrix and 0 is a null matrix.
- adj(BT) = adj(B)T, where BT is a matrix transpose from B.
- The product of a matrix B and its adjoint, which results in a diagonal matrix in which diagonal entries are the determinant det(B), is known as the adjoint. “I” is an identity matrix, and B adj(B) = adj(B) B = det(B) I.
- If C is a different square matrix, adj(BC) = adj(C) adj(B) will apply.
- adj(Bk) = adj(B)k for any non-negative integer k.
Applications of Adjoint of a Matrix
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Below are a few applications of a matrix adjoint:
- A matrix's adjoint aids in the solution of a system of linear equations.
- It enables us to determine whether an equation's solution is consistent or inconsistent.
- Models of physical structures are created by engineers and physicists, who also perform the precise calculations required to run challenging machinery.
- These matrices are essential for determining battery power outputs and for resistors' conversion of electricity into other forms of useful energy.
Also Read:
Things to Remember
- In a transposed matrix, the original matrix's columns become the transposed matrix's rows, and the original matrix's rows become the transposed matrix's columns.
- By transposing the cofactors matrix, the adjoint of a matrix is created.
- The determinant of the product matrices is the same as the product of the determinants.
- A matrix's adjoint aids in the solution of a system of linear equations.
- Each element in a square matrix B has a unique minor.
- Aij is the cofactor of the element aij, and the adjoint of a square matrix A = [aij]nxn is defined as the matrix [Aij]nxn transposed.
- After eliminating the row and column that correspond to that specific matrix element, the minor value can be calculated from the determinant of a square matrix.
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Sample Questions
Ques. What is the adjoint of a square matrix? (1 mark)
A) The transpose of the matrix
B) The inverse of the matrix
C) The determinant of the matrix
D) The identity matrix
Ans. D) The identity matrix
Explanation: The adjoint of a square matrix is a matrix denoted as adj(A) or sometimes A*, and it is found using the cofactor matrix of the original matrix A. The adjoint is used in various mathematical operations, such as finding the inverse of a matrix using the formula A-1 = (1/det(A)) * adj(A). Therefore, the correct answer is option D, the identity matrix is not the adjoint but a square matrix itself.
Ques. What is the relationship between the adjoint of a matrix and its eigenvalues? (1 mark)
A) The adjoint has the same eigenvalues as the original matrix.
B) The adjoint has the eigenvalues inverted (reciprocal) compared to the original matrix.
C) The adjoint has no eigenvalues.
D) The eigenvalues of the adjoint are always zero.
Ans. D) The eigenvalues of the adjoint are always zero.
Explanation: The eigenvalues of the adjoint are always zero. Eigenvalues of the adjoint are not necessarily zero for the original matrix.
Ques. Find the adjoint of the following 3x3 matrix A: (3 marks)
A = \(\begin{bmatrix}2 & 1 &3 \\[0.3em]0 & -1 & 2 \\[0.3em]1 & 0 &4 \\[0.3em] \end{bmatrix}\)
Ans. To find the adjoint of matrix A, we need to follow these steps:
Step 1: Find the matrix of minors.
Step 2: Calculate the matrix of cofactors.
Step 3: Transpose the matrix of cofactors to obtain the adjoint.
Step 1:
For each element of matrix A, we calculate the minor by removing the row and column containing that element and finding the determinant of the resulting 2x2 matrix.
Minor of A[1][1]:
|-1 2|
|0 4|
Minor(A[1][1]) = (-1) * (4) - (2) * (0) = -4
Minor of A[1][2]:
|0 2|
|1 4|
Minor(A[1][2]) = (0) * (4) - (2) * (1) = -2
Minor of A[1][3]:
|0 -1|
|1 0|
Minor(A[1][3]) = (0) * (0) - (-1) * (1) = 1
Continuing process for all elements of the matrix to obtain the matrix of minors:
Minors = \(\begin{bmatrix}-4 & 2 & 1 \\[0.3em]10 & -8 & 0 \\[0.3em]-1 & 0 & 2\\[0.3em] \end{bmatrix}\)
Step 2:To calculate the matrix of cofactors, we need to alternate the signs of the minors. If the element is in an odd position (i.e., not on an even row and column), we negate the minor.
Cofactors =\(\begin{bmatrix}-4 & 2 & 1 \\[0.3em]10 & -8 & 0 \\[0.3em]-1 & 0 & 2\\[0.3em] \end{bmatrix}\)
Step 3: Now, we transpose the matrix of cofactors to obtain the adjoint of matrix A:
Adjoint(A) = \(\begin{bmatrix}-4 & 10 & -1 \\[0.3em]2 & -8 & 0 \\[0.3em]1 & 0 & 2\\[0.3em] \end{bmatrix}\) So, the adjoint of matrix A is:
Adjoint(A) = \(\begin{bmatrix}-4 & 10 & -1 \\[0.3em]2 & -8 & 0 \\[0.3em]1 & 0 & 2\\[0.3em] \end{bmatrix}\)
Ques. When is the adjoint of a matrix orthogonal? (1 mark)
A) The adjoint is always orthogonal.
B) The adjoint is orthogonal if and only if the original matrix is orthogonal.
C) The adjoint is orthogonal if its determinant is 1.
D) The adjoint is orthogonal if its trace is 0.
Ans. B) The adjoint is orthogonal if and only if the original matrix is orthogonal.
Explanation: The adjoint is orthogonal if and only if the original matrix is orthogonal. Orthogonal matrices have the property that their inverse is equal to their transpose, and this relationship extends to their adjoints as well.
Ques. What possible arrangements may a matrix with 28 elements have? What if there are 13 components? (2 marks)
Ans. There are mn elements in a matrix of order m x n, where m and n are both natural numbers.
As you can see, m x n = 28.
(m, n) = {(1, 28), (2, 14), (4, 7), (7, 4), (14, 2), (28, 1)}
The feasible combinations are therefore 1 x 28, 2 x 14, 4 x 7, 7 x 4, and 14 x 2.
Additionally, if it contains 13 elements, m x n = 13.
(m, n) = {(1, 13), (13, 1)}
The possible orders are therefore 1 x 13, 13 x 1, and so on.
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