Question

What are the properties of matrix multiplication?

Answer 1

Hints Recall the formation of a matrix that is how a matrix looks like, then write the properties of matrix multiplication.

Complete step by step solution
Example of a matrix is \[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right]\] .
Now, suppose two matrices are \[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right]\]and \[\left[ {\begin{array}{*{20}{c}}p&q&r\\s&t&u\\v&w&x\end{array}} \right]\].
Hence, the multiplication of two matrices is,
\[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right].\left[ {\begin{array}{*{20}{c}}p&q&r\\s&t&u\\v&w&x\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ap + bs + cv}&{aq + bt + cw}&{ar + bu + cx}\\{dp + es + fv}&{dq + et + fw}&{dr + eu + fx}\\{gp + hs + iv}&{gq + ht + iw}&{gr + hu + ix}\end{array}} \right]\]

The properties of a matrix multiplication are,
(1) Associative law: If A, B and C are non-zero matrices then \[(AB)C = A(BC)\] .
(2) Distributive law: If A, B and C are non-zero matrices, then \[A.(B + C) = A.B + A.C\]
                                     Or, \[(B + C).A = B.A + C.A\].
(3) Multiplicative identity property: If I be the identity matrix then \[A.I = I.A = A\] .

The properties of a matrix multiplication are associative law, distributive law and the identity property.

Additional information:
The matrix multiplication does not follow commutative property. Suppose the order of A is \[m\times n\] and the order of B is \[i \times j\]. Then matrix multiplication of AB exists if i = n.

Note Sometime students give the proofs with the properties, but that is not needed as the question is asking only to mention the properties of matrix multiplication. Only you can do that show the matrix multiplication but not elaborately just state the multiplication and then go for the properties of matrix multiplication as per the requirement of the question.