# Symmetric matrix

A square matrix

*A*= [*a*] is called a symmetric matrix if_{ij}*a*=_{ij}*a*for all_{ji}*i*,*j*.For example, the matrix

*A*= is symmetric, because*a*_{12}= â€“1 =*a*_{21},*a*_{13}= 1 =*a*_{31},*a*_{23}= 5 =*a*_{32.}For symmetric matrix

*A*^{T}=*A*.# Skew-symmetric matrix

A square matrix

*A*= [*a*] is a skew-symmetric matrix if_{ij}*a*= â€“_{ij}*a*for all_{ji}*i*,*j*.For example, the matrix

*A*= is skew-symmetric.For skew-symmetric matrix

*A*^{T}= â€“*A.*# Adjoint of square matrix

Let

*A*= [*a*] be a square matrix of order_{ij}*n*and let*C*be cofactor of_{ij}*a*in_{ij}*A*. Then the transpose of the matrix of cofactors of elements of*A*is called the adjoint of*A*and is denoted by adj*A*.# Inverse of matrix

A non-singular square matrix of order

*n*is invertible if there exists a square matrix*B*of the same order such that*AB*=*I*=_{n}*BA*. In such a case, we say that the inverse of*A*is*B*and we write,*A*^{â€“1}=*B*. Also from*A*(adj*A*) = |*A*|*I*= (adj_{n}*A*)*A*, we can conclude that*A*^{â€“1}= â‹… adj*A*.# Properties of adjoint and inverse of a matrix

- Let
*A*be a square matrix of order*n*. Then*A*(adj*A*) = |*A*|*I*= (adj_{n}*A*)*A.* - Every invertible matrix possesses a unique inverse.
**Reversal law:**If*A*and*B*are invertible matrices of the same order, then*AB*is invertible and (*AB*)^{â€“1}=*B*^{â€“1}*A*^{â€“1}.*A*,*B*,*C*, ..., are invertible matrices then (*ABC*...)^{â€“1}= ...*C*^{â€“1}*B*^{â€“1}*A*^{â€“1}.- If
*A*is an invertible square matrix, then*A*^{T}is also invertible and (*A*^{T})^{â€“1}= (*A*^{â€“1})^{T}. - If
*A*is a non-singular square matrix of order*n*. Then |adj*A*| = |*A*|^{n}^{â€“1}. **Reversal law for adjoint:**If*A*and*B*are non-singular square matrices of the same order, then adj â‹… (*AB*) = (adj*B*) (adj*A*).- If
*A*is an invertible square matrix, then adj(*A*^{T}) = (adj*A*)^{T}. - If
*A*is a non-singular square matrix, then adj(adj*A*) = |*A*|^{n}^{â€“2}*A.* - If
*A*is a non-singular matrix, then prove that |*A*^{â€“1}| = |*A*|^{â€“1}, i.e., |*A*^{â€“1}| = . - The inverse of the
*k*th power of*A*is the*k*th power of the inverse of*A.*

Consider the equations

*a*

_{1}

*x*+

*b*

_{1}

*y*+

*c*

_{1}

*z*=

*d*

_{1}

*a*

_{2}

*x*+

*b*

_{2}

*y*+

*c*

_{2}

*z*=

*d*

_{2 ...}(1)

*a*

_{3}

*x*+

*b*

_{3}

*y*+

*c*

_{3}

*z*=

*d*

_{3}

If

Then (1) is equivalent to the matrix equation

*AX*=

*D*...(2)

Multiplying both sides of (2) by the reciprocal matrix

*A*^{â€“1}, we get*A*

^{â€“1}(

*AX*) =

*A*-

^{â€“1}

*D*â‡’

*IX*=

*A*

^{â€“1 }

*D*[âˆµ

*A*

^{â€“1}

*A*= 1]

â‡’

*X*=*A*^{â€“1 }*D*.- If
*A*is a non-singular matrix, then the system of equations given by*AX*=*B*has a unique solution given by*X*=*A*^{â€“1}*B* - If
*A*is singular matrix, and (*ajd**A*)*D*= O, then the system of the equations given by*AX*=*D*is consistent with infinitely many solutions. - If
*A*is singular matrix, and (adj A)*D*â‰*O*, then the system of equation given by*AX*=*D*is inconsistent.