If [(1, 0, 0), (0, -1, 0), (0, 0, -1)][x, y, z] = [1, 0, 1], find x, y and z.

Question

If  \(\begin{bmatrix} 1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}\begin{bmatrix} x\\y\\z\end{bmatrix} \)= \(\begin{bmatrix} 1\\ 0\\1\end{bmatrix},\) find x, y and z.

Answer 1

The given matrix equation is :-

  \(\begin{bmatrix} 1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}\begin{bmatrix} x\\y\\z\end{bmatrix} \)= \(\begin{bmatrix} 1\\ 0\\1\end{bmatrix}\)

Let A = \(\begin{bmatrix} 1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}\); P = \(\begin{bmatrix} x\\y\\z\end{bmatrix} \) and B = \(\begin{bmatrix} 1\\ 0\\1\end{bmatrix}\)

So, we can write the equation as

AP = B

Pre-multiplying A^-1 both sides we get,

A^-1AP = A^-1B

IP = A^-1B ( \(\because\) A ^-1A = I )

P = A^-1B ( IP = P ) …….(i)

Now,