Question

Check whether the multiplication of a matrix is associative.

Answer 1

Hint: Matrix is a table of numbers, symbols, arranged in rows and columns. The dimension of a matrix is represented as rows multiplied with columns. For example, $\left( {2 \times 3} \right)$, here are two rows and three columns. When a rectangular array is formed by arranging the numbers in rows and columns is termed as matrix.

Complete step by step solution:
We have to check whether the matrix multiplication follows associative property or not.
For any three given matrices $A$, $B$ and $C$, the associative property is given as,
$\left( {A \times B} \right) \times C = A \times \left( {B \times C} \right)$
We can assume three matrices with general terms as,
\[
  A = \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d \\
  e&f
\end{array}} \right] \\
  B = \left[ {\begin{array}{*{20}{c}}
  g&h&i&j \\
  k&l&m&n
\end{array}} \right] \\
  C = \left[ {\begin{array}{*{20}{c}}
  o&p \\
  q&r \\
  s&t \\
  u&v
\end{array}} \right] \\
 \]
While assuming the order of the matrices we have to make sure that calculation of both $\left( {A \times B} \right) \times C$ and $A \times \left( {B \times C} \right)$ is possible.
Now first we evaluate the LHS, i.e. $\left( {A \times B} \right) \times C$.
$\left( {A \times B} \right) \times C = \left( {\left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d \\
  e&f
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  g&h&i&j \\
  k&l&m&n
\end{array}} \right]} \right) \times \left[ {\begin{array}{*{20}{c}}
  o&p \\
  q&r \\
  s&t \\
  u&v
\end{array}} \right]$
We can simplify this as,
\[
  \left( {\left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d \\
  e&f
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  g&h&i&j \\
  k&l&m&n
\end{array}} \right]} \right) \times \left[ {\begin{array}{*{20}{c}}
  o&p \\
  q&r \\
  s&t \\
  u&v
\end{array}} \right] \\
   = \left[ {\begin{array}{*{20}{c}}
  {ag + bk}&{ah + bl}&{ai + bm}&{aj + bn} \\
  {cg + dk}&{ch + dl}&{ci + dm}&{cj + dn} \\
  {ej + fk}&{eh + fl}&{ei + fm}&{ej + fn}
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  o&p \\
  q&r \\
  s&t \\
  u&v
\end{array}} \right] \\
   = \left[ {\begin{array}{*{20}{c}}
  {\left( {ag + bk} \right)o + \left( {ah + bl} \right)q + \left( {ai + bm} \right)s + \left( {aj + bn} \right)u}&{\left( {ag + bk} \right)p + \left( {ah + bl} \right)r + \left( {ai + bm} \right)t + \left( {aj + bn} \right)v} \\
  {\left( {cg + dk} \right)o + \left( {ch + dl} \right)q + \left( {ci + dm} \right)s + \left( {cj + dn} \right)u}&{\left( {cg + dk} \right)p + \left( {ch + dl} \right)r + \left( {ci + dm} \right)t + \left( {cj + dn} \right)v} \\
  {\left( {ej + fk} \right)o + \left( {eh + fl} \right)q + \left( {ei + fm} \right)s + \left( {ej + fn} \right)u}&{\left( {ej + fk} \right)p + \left( {eh + fl} \right)r + \left( {ei + fm} \right)t + \left( {ej + fn} \right)v}
\end{array}} \right] \\
   = \left[ {\begin{array}{*{20}{c}}
  {ago + bko + ahq + blq + ais + bms + aju + bnu}&{agp + bkp + ahr + blr + ait + bmt + ajv + bnv} \\
  {cgo + dko + chq + dlq + cis + dms + cju + dnu}&{cgp + dkp + chr + dlr + cit + dmt + cjv + dnv} \\
  {ejo + fko + ehq + flq + eis + fms + eju + fnu}&{ejp + fkp + ehr + flr + eit + fmt + ejv + fnv}
\end{array}} \right] \\
 \]
Now we will evaluate the RHS, i.e. $A \times \left( {B \times C} \right)$.
$A \times \left( {B \times C} \right) = \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d \\
  e&f
\end{array}} \right] \times \left( {\left[ {\begin{array}{*{20}{c}}
  g&h&i&j \\
  k&l&m&n
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  o&p \\
  q&r \\
  s&t \\
  u&v
\end{array}} \right]} \right)$
We can simplify this as,
\[
  \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d \\
  e&f
\end{array}} \right] \times \left( {\left[ {\begin{array}{*{20}{c}}
  g&h&i&j \\
  k&l&m&n
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  o&p \\
  q&r \\
  s&t \\
  u&v
\end{array}} \right]} \right) \\
   = \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d \\
  e&f
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  {go + hq + is + ju}&{gp + hr + it + jv} \\
  {ko + lq + ms + nu}&{kp + lr + mt + nv}
\end{array}} \right] \\
   = \left[ {\begin{array}{*{20}{c}}
  {a\left( {go + hq + is + ju} \right) + b\left( {ko + lq + ms + nu} \right)}&{a\left( {gp + hr + it + jv} \right) + b\left( {kp + lr + mt + nv} \right)} \\
  {c\left( {go + hq + is + ju} \right) + d\left( {ko + lq + ms + nu} \right)}&{c\left( {gp + hr + it + jv} \right) + d\left( {kp + lr + mt + nv} \right)} \\
  {e\left( {go + hq + is + ju} \right) + f\left( {ko + lq + ms + nu} \right)}&{e\left( {gp + hr + it + jv} \right) + f\left( {kp + lr + mt + nv} \right)}
\end{array}} \right] \\
   = \left[ {\begin{array}{*{20}{c}}
  {ago + ahq + ais + aju + bko + blq + bms + bnu}&{agp + ahr + ait + ajv + bkp + blr + bmt + bnv} \\
  {cgo + chq + cis + cju + dko + dlq + dms + dnu}&{cgp + chr + cit + cjv + dkp + dlr + dmt + dnv} \\
  {ejo + ehq + eis + eju + fko + flq + fms + fnu}&{ejp + ehr + eit + ejv + fkp + flr + fmt + fnv}
\end{array}} \right] \\
   = \left[ {\begin{array}{*{20}{c}}
  {ago + bko + ahq + blq + ais + bms + aju + bnu}&{agp + bkp + ahr + blr + ait + bmt + ajv + bnv} \\
  {cgo + dko + chq + dlq + cis + dms + cju + dnu}&{cgp + dkp + chr + dlr + cit + dmt + cjv + dnv} \\
  {ejo + fko + ehq + flq + eis + fms + eju + fnu}&{ejp + fkp + ehr + flr + eit + fmt + ejv + fnv}
\end{array}} \right] \\
 \]
We can observe that we got $\left( {A \times B} \right) \times C = A \times \left( {B \times C} \right)$.
Hence we can say that matrix multiplication is associative.

Note: In this problem, we have used the associative property, which says the multiplication of any three matrices is same,$\left( {A \times B} \right) \times C = A \times \left( {B \times C} \right)$ , either we multiply first two matrices and then multiply it with third or we multiply last two matrices and then multiply it with the first.