DETERMINANT OF A MATRIX, ADJOINT OF A MATRIX, SINGULAR MATRIX, NON SINGULAR MATRIX

Determinant of a matrix, Adjoint of a matrix, Singular matrix, Non Singular matrix

Determinant of a matrix, Adjoint of a matrix, Singular matrix, Non Singular matrix

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Determinant of a Matrix


The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix and can be used in various applications like solving linear equations, finding the inverse of a matrix, and in calculus for change of variables in integrals.

For a 2x2 matrix A=(abcd)A = begin{pmatrix} a & b c & d end{pmatrix}A=(ac​bd​), the determinant is calculated as:

det(A)=ad−bctext{det}(A) = ad - bcdet(A)=ad−bc

For a 3x3 matrix A=(abcdefghi)A = begin{pmatrix} a & b & c d & e & f g & h & i end{pmatrix}A=​adg​beh​cfi​​, the determinant is:

det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)

The determinant can be computed for larger matrices using various methods, including cofactor expansion and row reduction.

Adjoint of a Matrix


The adjoint (or adjugate) of a matrix is the transpose of the cofactor matrix. For a given square matrix AAA, the cofactor CijC_{ij}Cij​ is calculated by:

 

  • Removing the i-th row and j-th column from AAA.



  • Calculating the determinant of the resulting (n−1)×(n−1)(n-1) times (n-1)(n−1)×(n−1) matrix.



  • Multiplying the determinant by (−1)i+j(-1)^{i+j}(−1)i+j.


 

 The adjoint of AAA is then the transpose of the matrix of cofactors. For a 3x3 matrix A=(abcdefghi)A = begin{pmatrix} a & b & c d & e & f g & h & i end{pmatrix}A=​adg​beh​cfi​​, its adjoint adj(A)text{adj}(A)adj(A) is:

adj(A)=(C11C12C13C21C22C23C31C32C33)Ttext{adj}(A) = begin{pmatrix} text{C}_{11} & text{C}_{12} & text{C}_{13} text{C}_{21} & text{C}_{22} & text{C}_{23} text{C}_{31} & text{C}_{32} & text{C}_{33} end{pmatrix}^Tadj(A)=​C11​C21​C31​​C12​C22​C32​​C13​C23​C33​​​T

Singular Matrix


A matrix is called singular if its determinant is zero. Singular matrices do not have an inverse. This property indicates that the matrix represents a linear transformation that squashes the n-dimensional space into a lower dimension, leading to a loss of information.

For example, if det(A)=0text{det}(A) = 0det(A)=0, then matrix AAA is singular.

Non-Singular Matrix


A matrix is called non-singular if its determinant is non-zero. Non-singular matrices have an inverse. This property indicates that the matrix represents a linear transformation that is bijective (one-to-one and onto), meaning no information is lost, and the transformation is reversible.

For example, if det(A)≠0text{det}(A) neq 0det(A)=0, then matrix AAA is non-singular.

Summary:


 Determinant: A scalar value indicating specific properties of a square matrix.

Adjoint: The transpose of the cofactor matrix of a given square matrix.


 Singular Matrix: A matrix with a determinant of zero, having no inverse.

 Non-Singular Matrix: A matrix with a non-zero determinant, having an inverse.

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