Matrices EAMCET – Part 1

Here are some of the most important points from matrices to be understood and remembered by EAMCET aspirants.

  1. Product of two upper triangular matrices is a upper triangular matrix.
  2. Determinant of triangular matrix is the product of the elements in the principle diagonal.
  3. (i) trace (KA) = K tr(A)
    (ii) tr (A + B) = tr(A) + tr(B)
    (iii) tr (AB) = tr (BA)
  4. If A is a skew symmetric matrix then,
    1-> All the elements in the principle diagonal are equal to zero.
    2-> |A|≠ 0 then its order is an odd number.
  5. If A, B are symmetric matrices, K is a scalar then
    (i) KA is symmetric.
    (ii) (A + B) and (A – B) are symmetric.
    (iii) AB, BA are symmetric if and only if AB = BA.
  6. For any square matrix A, (A + AT) is symmetric and (A – AT) is skew symmetric.
  7. A is a matrix of order m x n then A.AT, AT.A are symmetric matrices.
    A is matrix of order (m x n)
  8. The only non-singular idempotent matrix is the unit matrix. That is if A2 = A and A ≠ I then |A| = 0
  9. If A, B are square matrices of the same type satisfying the condition AB = A, BA = B then
    (i) Both A and B are idempotent matrices
    (ii) For any natural number ‘n’, (A+B)n = 2(n-1) (A+B)
    for any natural number n
  10. If A is an involutary matrix then,
    If A is involutary matrix
  11. If A is an orthogonal matrix then,
    A is orthogonal matrix

    Determinant of A is ±1. If the rays are in right handed system then the value of determinant is +1 and if they are in left handed system then the value is -1.

  12. Every nilpotent matrix is a singular matrix.
  13. If A is a nilpotent matrix of index 3 then,
    A is nilpotent matrix
  14. If A is involutary matrix and A ≠ I, then the determinant of |I + A| = 0.
    Proof :-
    A is involutary and not equal to I