Matrix

Classical Adjoint (Adjugate) Matrix

Cofactor Formula

A Cofactor, in mathematics, is used to find the inverse of the matrix, adjoined. The Cofactor is the number you get when you remove the column and row of a designated element in a matrix, which is just a numerical grid in the form of rectangle or a square. The cofactor is always preceded by a positive (+) or negative (-) sign. Let \(\mathbf{A}\) be an \(n\times n\) matrix and let \(M_{ij}\) be the \((n-1)\times (n-1)\) matrix obtained by deleting the \(i^{th}\) row and \(j^{th}\) column. Then, \(\text{det}M_{ij}\) is called the minor of \(a_{ij}\). The cofactor \(A_{ij}`of :math:`a_{ij}\) is defined by:

\[A_{ij}=(-1)^{i+j}\text{det}M_{ij}\]

Minors and Cofactors

• Minors and Cofactors

• Adjoint Of a Matrix

What Are Minors?

The minor of an element in a matrix is defined as the determinant obtained by deleting the row and column in which that element lies. For example, in the determinant

\[\begin{split}D=\begin{vmatrix} a_{11}& a_{12} & a_{13}\\ a_{21}& a_{22} & a_{23}\\ a_{31}& a_{32} & a_{33}\\ \end{vmatrix}\end{split}\]

minor of \(a_{12}\) is denoted as \(M_{12}\). Here,

\[\begin{split}M_{12}=\begin{vmatrix} a_{21}& a_{23}\\ a_{31}& a_{33}\\ \end{vmatrix}\end{split}\]

What Are Cofactors?

Cofactor of an element aij is related to its minor as

\[C_{ij}=(-1)^{i+j}M_{ij}\]

where \(i\) denotes the \(i^{th}\) row and \(j\) denotes the \(i^{jth}\) column to which the element \(a_{ij}\) belongs. Now, we define the value of the determinant of order three in terms of ‘Minor’ and ‘Cofactor’ as

\[D=a_{11}M_{11}-a_{12}M_{12}+a_{13}M_{13}\]

\[D=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\]

The classical adjoint matrix should not be confused with the adjoint matrix. The adjoint is the conjugate transpose of a matrix while the classical adjoint is another name for the adjugate matrix or cofactor transpose of a matrix.

\[\begin{split}\mathbf{A}^{*}=\begin{bmatrix} A_{11}&A_{21} &\cdots & A_{n1}\\ A_{12}&A_{22} &\cdots & A_{n2}\\ \vdots& \vdots & &\vdots \\ A_{1n}&A_{2n} &\cdots & A_{nn}\\ \end{bmatrix}\end{split}\]

Transpose

Transpose

Formally, the \(i\)-th row, \(j\)-th column element of \(\mathbf{A}^{\text{T}}\) is the \(j\)-th row, \(i\)-th column element of \(\mathbf{A}\):

\[[\mathbf{A}^{\text{T}}]_{ij}=[\mathbf{A}]_{ji}\]

Properties

Let \(\mathbf{A}\) and \(\mathbf{B}\) be matrices and \(c\) be a scalar.

\[{\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\mathbf {A} .}\]

\[{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }.}\]

\[{\displaystyle \left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }.}\]

What are Eigenvalues?

The eigenvalue is explained to be a scalar associated with a linear set of equations which, when multiplied by a nonzero vector, equals to the vector obtained by transformation operating on the vector.

Let us consider \(k \times k\) square matrix \(A\) and \(\mathbf{v}\) be a vector, then \(\lambda\) is a scalar quantity represented in the following way:

\[A\mathbf{v} = \lambda\mathbf{v}\]

Here, \(\lambda\) is considered to be the eigenvalue of matrix \(A\).

The above equation can also be written as:

\[(A – \lambda I) = 0\]

Where “\(I\)” is the identity matrix of the same order as \(A\).

This equation can be represented in the determinant of matrix form.

\[|A – \lambda I| = 0\]

The above relation enables us to calculate eigenvalues \(\lambda\) easily.