# Symmetric matrix

A square matrix A = [aij] is called a symmetric matrix if aij = aji for all i, j.

For example, the matrix A = is symmetric, because a12 = â€“1 = a21, a13 = 1 = a31, a23 = 5 = a32.

For symmetric matrix AT = A.

# Skew-symmetric matrix

A square matrix A = [aij] is a skew-symmetric matrix if aij = â€“aji for all i, j.

For example, the matrix A = is skew-symmetric.

For skew-symmetric matrix AT = â€“A.

Let A = [aij] be a square matrix of order n and let Cij be cofactor of aij in 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 = In = BA. In such a case, we say that the inverse of A is B and we write, Aâ€“1 = B. Also from A(adj A) = |A| In = (adj A) A, we can conclude that Aâ€“1 = â‹… adj A.

# Properties of adjoint and inverse of a matrix

1. Let A be a square matrix of order n. Then A(adj A) = |A| In = (adj A) A.
2. Every invertible matrix possesses a unique inverse.
3. Reversal law: If A and B are invertible matrices of the same order, then AB is invertible and (AB)â€“1 = Bâ€“1Aâ€“1.

In general, if A, B, C, ..., are invertible matrices then (ABC ...)â€“1 = ... Câ€“1Bâ€“1Aâ€“1.
4. If A is an invertible square matrix, then AT is also invertible and (AT)â€“1 = (Aâ€“1)T.
5. If A is a non-singular square matrix of order n. Then |adj A| = |A|nâ€“1.
6. 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).
7. If A is an invertible square matrix, then adj(AT) = (adj A)T.
8. If A is a non-singular square matrix, then adj(adj A) = |A|nâ€“2 A.
9. If A is a non-singular matrix, then prove that |Aâ€“1| = |A|â€“1, i.e., |Aâ€“1| = .
10. The inverse of the kth power of A is the kth power of the inverse of A.
Consider the equations

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2   ...(1)
a3x + b3y + c3z = d3

If
Then (1) is equivalent to the matrix equation

AX = D ...(2)

Multiplying both sides of (2) by the reciprocal matrix Aâ€“1, we get

Aâ€“1 (AX) = A-â€“1D â‡’ IX = Aâ€“1 D [âˆµ Aâ€“1 A = 1]
â‡’ X = Aâ€“1 D.
1. If A is a non-singular matrix, then the system of equations given by AX = B has a unique solution given by X = Aâ€“1B
2. 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.
3. If A is singular matrix, and (adj A) D â‰  O, then the system of equation given by AX = D is inconsistent.