# JEE Main & Advanced Mathematics Determinants & Matrices Multiplication of Matrices

Multiplication of Matrices

Category : JEE Main & Advanced

Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post multiplier). Thus, if $A={{[{{a}_{ij}}]}_{m\times n}}$ and $B={{[{{b}_{ij}}]}_{n\times p}}$ are two matrices of order $m\times n$ and $n\times p$respectively, then their product AB is of order $m\times p$and is defined as ${{(AB)}_{ij}}=\sum\limits_{r=1}^{n}{{{a}_{ir}}{{b}_{rj}}}$=[{{a}_{i1}}{{a}_{i2}}...{{a}_{in}}]\left[ \begin{align} & \underset{\vdots }{\mathop{\overset{{{b}_{1j}}}{\mathop{{{b}_{2j}}}}\,}}\, \\ & {{b}_{nj}} \\ \end{align} \right]= (${{i}^{th}}$ row of A)(${{j}^{th}}$ column of B)                                                                                                            .....(i)

where $i=1,\text{ }2,\text{ }...,m$ and $j=1,\text{ }2,\text{ }...p$

Now we define the product of a row matrix and a column matrix.

Let $A=\left[ {{a}_{1}}{{a}_{2}}....{{a}_{n}} \right]$be a row matrix and $B=\left[ \begin{matrix} {{b}_{1}} \\ \underset{\vdots }{\mathop{{{b}_{2}}}}\, \\ {{b}_{n}} \\ \end{matrix} \right]$ be a column matrix.

Then $AB=\left[ {{a}_{1}}{{b}_{1}}+{{a}_{2}}{{b}_{2}}+....+{{a}_{n}}{{b}_{n}} \right]$                             ?..(ii)

Thus, from (i), ${{(AB)}_{ij}}=$Sum of the product of elements of ${{i}^{th}}$ row of A with the corresponding elements of ${{j}^{th}}$ column of B.

Properties of matrix multiplication

If A, B and C are three matrices such that their product is defined, then

(i) $AB\ne BA$,           (Generally not commutative)

(ii) $(AB)C=A(BC)$,          (Associative Law)

(iii) $IA=A=AI$, where I is identity matrix for matrix multiplication.

(iv) $A(B+C)=AB+AC$, (Distributive law)

(v)  If $AB=AC\not{\Rightarrow }B=C$,(Cancellation law is not applicable)

(vi) If $AB=0,$ it does not mean that $A=0$ or $B=0,$ again product of two non zero matrix may be a zero matrix.

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