Lecture on Multiplication
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Multiplication of Matrix with a scalar:

\(k\begin{vmatrix}a&b\\c&d\end{vmatrix} = \begin{vmatrix}ka&kb\\kc&kd\end{vmatrix} \)
Example:
\(2*\begin{vmatrix}7&3\\9&1\end{vmatrix} = \begin{vmatrix}14&6\\18&2 \end{vmatrix} \)

Multiplication of Matrix with with another matrix:
M=Aij X Bpq is possible only when j = p and order of M is Mjk

Properties of Matrix Multiplication:

  1. k(A + B) = kA +kB
  2. p(qA) = (pq)A = q(pA)
  3. -p(A) = -(pA) = p(-A)
  • Important points about matrix multiplication:
  1. Not commutative
    AB ≠ BA
  2. Associative
    A(BC) = (AB)C
  3. Distributive over addition 
    A(B+C) = AB + AC
  4. AB = 0  doesn't mean  A = 0 or B = 0
  5. AB = AC ⇒ B = C iff A is non-singular matrix.
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