JEE Main & Advanced Mathematics Determinants & Matrices Multiplication of Matrices

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.