Name: Adding and Subtracting Matrices
Uploaded: 2019-07-19
Duration: 6 min 12 s
Description: Two matrices with the same order can be added or subtracted. Watch this video to learn some examples based on the addition and subtraction of matrices. Let's...

Question

Show that the following matrices are commutative under matrix multiplication

$A_{i} = (\begin{array}{cc} \cos θ_{i} & - \sin θ_{i} \\ \sin θ_{i} & \cos θ_{i} \end{array})$ , $i = 1, 2$

Accepted Answer

Given that $A_{i} = (\begin{array}{cc} \cos θ_{i} & - \sin θ_{i} \\ \sin θ_{i} & \cos θ_{i} \end{array})$

$A_{1} = (\begin{array}{cc} \cos θ_{1} & - \sin θ_{1} \\ \sin θ_{1} & \cos θ_{1} \end{array}), A_{2} = (\begin{array}{cc} \cos θ_{2} & - \sin θ_{2} \\ \sin θ_{2} & \cos θ_{2} \end{array})$

We know that multiplication of matrices $(A B)$ is possible when number of columns of $A$ is equal to number of rows of $B$ .

Condition for commutative property of multiplication is $A B = B A$ .

$A_{i} = (\begin{array}{cc} \cos θ_{i} & - \sin θ_{i} \\ \sin θ_{i} & \cos θ_{i} \end{array})$

$= (\begin{matrix} \cos (θ_{1} + θ_{2}) & - \sin (θ_{1} + θ_{2}) \\ \sin (θ_{1} - θ_{2}) & \cos (θ_{1} + θ_{2}) \end{matrix})$

$A_{2} A_{1} = (\begin{array}{cc} \cos θ_{2} & - \sin θ_{2} \\ \sin θ_{2} & \cos θ_{2} \end{array}) (\begin{array}{cc} \cos θ_{1} & - \sin θ_{1} \\ \sin θ_{1} & \cos θ_{1} \end{array}) = (\begin{matrix} \cos θ_{2} \cos θ_{1} - \sin θ_{2} \sin θ_{1} & - \cos θ_{2} \sin θ_{1} - \sin θ_{2} \cos θ_{1} \\ \sin θ_{2} \cos θ_{1} - \cos θ_{2} \sin θ_{1} & - \sin θ_{2} \sin θ_{1} + \cos θ_{2} \cos θ_{1} \end{matrix})$

$= (\begin{matrix} \cos (θ_{2} + θ_{1}) & - \sin (θ_{2} + θ_{1}) \\ \sin (θ_{2} - θ_{1}) & \cos (θ_{2} + θ_{1}) \end{matrix})$

$∴ A_{1} A_{2} = A_{2} A_{1}$ .

Show that the following matrices are commutative under matrix multiplication

$A_{i} = (\begin{array}{cc} \cos θ_{i} & - \sin θ_{i} \\ \sin θ_{i} & \cos θ_{i} \end{array})$ , $i = 1, 2$

Important Questions on Elementary Theory of Matrices

Learn and Prepare for any exam you want !

Show that the following matrices are commutative under matrix multiplication Ai=cos θi−sin θisin θicos θi, i=1, 2

Important Questions on Elementary Theory of Matrices

Learn and Prepare for any exam you want !

Show that the following matrices are commutative under matrix multiplication

$A_{i} = (\begin{array}{cc} \cos θ_{i} & - \sin θ_{i} \\ \sin θ_{i} & \cos θ_{i} \end{array})$ , $i = 1, 2$