Identity matrix “ construct “

Question

I'm not sure if the tags I added are the right ones ... correct me if I'm wrong name of course I study is Computer Programming.

I have solved all the homework problems. This is the last question I didn't understand. Construct the matrix

$$B=\pmatrix{ 1 & 2 & 3\\ 4 & 2 & 6\\ -3 & -1 & -3\\ }$$

Show that $B \times B \times B$ is a scalar multiple of the identity matrix and find the scalar.

I don't need you to give me an answer, help me understand how to solve it. Thanks

Edit: I am sorry I think I didn't make it clear .. I am solving the homework on R program.. how to write the answer? How can I enter the matrix on the program so I can multiply it ?

Answer 1

You just have to multiply B by itself 3 times. You will find the result to be a diagonal Matrix equal to a scalar multiplied by the identity matrix.

So basically you need to find a scalar $a \in \mathbb{R}$ which satisfies

$B(BB) = \begin{pmatrix}a&0&0\\0&a&0\\0&0&a\\ \end{pmatrix} $

Answer 2

$$B = \begin{pmatrix} 1&2&3\\4&2&6\\-3&-1&-3\end{pmatrix}$$

$$B\times B = \begin{pmatrix} 0&3&6\\-6&6&6\\2&-5&-6\end{pmatrix}$$

$$B\times B \times B = \begin{pmatrix} -6&0&0\\0&-6&0\\0&0&-6\end{pmatrix}= -6 I$$

Answer 3

$$B^2=B \times B=\pmatrix{ 1 & 2 & 3\\ 4 & 2 & 6\\ -3 & -1 & -3\\ }\pmatrix{ 1 & 2 & 3\\ 4 & 2 & 6\\ -3 & -1 & -3\\ }=\pmatrix{ 0 & 3 & 6\\ -6 & 6 & 6\\ 2 & -5 & -6\\ }$$ $$B \times B^2=\pmatrix{ 1 & 2 & 3\\ 4 & 2 & 6\\ -3 & -1 & -3\\ }\pmatrix{ 0 & 3 & 6\\ -6 & 6 & 6\\ 2 & -5 & -6\\ }=\pmatrix{ -6 & 0 & 0\\ 0& -6 & 0\\ 0 & 0 & -6\\}=-6 \times I_3$$