Name: Matrix Multiplication and Associated Properties
Uploaded: 2019-07-19
Duration: 6 min 22 s
Description: We learned the addition and subtraction of matrices. Is it possible to multiply the two matrices? Watch this video and learn to multiply two matrices together.

Question

If $M = \{[\begin{matrix} x & x \\ x & x \end{matrix}], x \neq 0, x \in R\}$ and $*$ be an operation on $M$ defined by $A * B = AB$ , for all $A, B \in M$ .

Prove that the operation $*$ is commutative as well as associative.

Accepted Answer

It is given that,

$M = \{[\begin{matrix} x & x \\ x & x \end{matrix}], x \neq 0, x \in R\}$ and $M$ be an operation on $A * B = AB$ defined by $A * B = AB$ , for all $A, B \in M$

To be proven: operation $*$ is commutative as well as associative

Now,

$\forall A, B \in M,$ Let $A = [\begin{matrix} x & x \\ x & x \end{matrix}]$ and $B = [\begin{matrix} y & y \\ y & y \end{matrix}]$ where $x, y \in R$ , $x \neq 0, y \neq 0$

Commutative $: A * B = AB = [\begin{matrix} x & x \\ x & x \end{matrix}] [\begin{matrix} y & y \\ y & y \end{matrix}] = [\begin{matrix} x y + x y & x y + x y \\ x y + x y & x y + x y \end{matrix}]$

$= [\begin{matrix} 2 x y & 2 x y \\ 2 x y & 2 x y \end{matrix}]$

And $B * A = BA = [\begin{matrix} y & y \\ y & y \end{matrix}] [\begin{matrix} x & x \\ x & x \end{matrix}] = [\begin{matrix} y x + y x & y x + y x \\ y x + y x & y x + y x \end{matrix}]$

$= [\begin{matrix} 2 x y & 2 x y \\ 2 x y & 2 x y \end{matrix}]$

Thus, $A * B = B * A$

Hence, $*$ is commutative on $M$ .

Let $M_{_{1}}, M_{2} a n d M_{3} b e t h r e e m a t r i c e s i d e n t i c a l t o M .$

Let $A = [\begin{matrix} x & x \\ x & x \end{matrix}], B = [\begin{matrix} y & y \\ y & y \end{matrix}]$ and $C = [\begin{matrix} z & z \\ z & z \end{matrix}]$

Where $x, y, z \in R, x \neq 0, y \neq 0, z \neq 0$

Here, $(A * B) * C = (AB) * C = (AB) C$

$= ([\begin{matrix} x & x \\ x & x \end{matrix}] [\begin{matrix} y & y \\ y & y \end{matrix}]) [\begin{matrix} z & z \\ z & z \end{matrix}] = [\begin{matrix} x y + x y & x y + x y \\ x y + x y & x y + x y \end{matrix}] [\begin{matrix} z & z \\ z & z \end{matrix}]$

$= [\begin{matrix} 2 x y & 2 x y \\ 2 x y & 2 x y \end{matrix}] [\begin{matrix} z & z \\ z & z \end{matrix}] = [\begin{matrix} 2 x y z & 2 x y z \\ 2 x y z & 2 x y z \end{matrix}]$

And $A * (B * C) = A * (BC) = A (BC)$

$= [\begin{matrix} x & x \\ x & x \end{matrix}] ([\begin{matrix} y & y \\ y & y \end{matrix}] [\begin{matrix} z & z \\ z & z \end{matrix}]) = [\begin{matrix} x & x \\ x & x \end{matrix}] [\begin{matrix} y z + y z & y z + y z \\ y z + y z & y z + y z \end{matrix}]$

$= [\begin{matrix} x & x \\ x & x \end{matrix}] [\begin{matrix} 2 y z & 2 y z \\ 2 y z & 2 y z \end{matrix}] = [\begin{matrix} 2 x y z & 2 x y z \\ 2 x y z & 2 x y z \end{matrix}]$ [ $∵ x, y, z \in R$ and associative law holds under multiplication]

Thus, $(A * B) * C = A * (B * C)$

Hence, $*$ is associative on $M$ .

If $M = \{[\begin{matrix} x & x \\ x & x \end{matrix}], x \neq 0, x \in R\}$ and $$ be an operation on $M$ defined by $A B = AB$ , for all $A, B \in M$ .

Prove that the operation $*$ is commutative as well as associative.

Important Questions on Matrices

Learn and Prepare for any exam you want !

If M=xxxx, x≠0, x∈R and * be an operation on M defined by A*B=AB, for all A,B∈M. Prove that the operation * is commutative as well as associative.

Important Questions on Matrices

Learn and Prepare for any exam you want !

If $M = \{[\begin{matrix} x & x \\ x & x \end{matrix}], x \neq 0, x \in R\}$ and $$ be an operation on $M$ defined by $A B = AB$ , for all $A, B \in M$ .

Prove that the operation $*$ is commutative as well as associative.