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Symmetric matrix

A square matrix A = [aij] is called a symmetric matrix if aij = aji for all i, j.
 
For example, the matrix A = 63275.png is symmetric, because a12 = –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 = –aji for all i, j.
 
For example, the matrix A = 63269.png is skew-symmetric.
 
For skew-symmetric matrix AT = –A.

Adjoint of square matrix

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, A–1 = B. Also from A(adj A) = |A| In = (adj A) A, we can conclude that A–1 = 63262.png 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)–1 = B–1A–1.
     
    In general, if A, B, C, ..., are invertible matrices then (ABC ...)–1 = ... C–1B–1A–1.
  4. If A is an invertible square matrix, then AT is also invertible and (AT)–1 = (A–1)T.
  5. If A is a non-singular square matrix of order n. Then |adj A| = |A|n–1.
  6. Reversal law for adjoint: If A and B are non-singular square matrices of the same order, then adj (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|n–2 A.
  9. If A is a non-singular matrix, then prove that |A–1| = |A|–1, i.e., |A–1| = 63256.png.
  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 64576.png
Then (1) is equivalent to the matrix equation
 
AX = D ...(2)
 
Multiplying both sides of (2) by the reciprocal matrix A–1, we get
 
A–1 (AX) = A-–1D IX = A–1 D [ A–1 A = 1]
X = A–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 = A–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) DO, then the system of equation given by AX = D is inconsistent.




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