Question

Consider the matrices

$\begin{array}{l}\mathbf{C}\mathbf{=}\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]\mathbf{,}\mathbf{H}\mathbf{=}\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 1\\ 1& 0& 1\end{array}\right]\mathbf{,}\\ \mathbf{L}\mathbf{=}\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]\mathbf{,}\mathbf{T}\mathbf{=}\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right]\mathbf{,}\\ \mathbf{X}\mathbf{=}\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]\mathbf{,}\mathbf{Y}\mathbf{=}\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right]\end{array}$

1. Which of the matrices in this list have the same kernel as matrix $\mathbf{C}$ ?
2. Which of the matrices in this list have the same image as matrix $\mathbf{C}$?
3. Which of these matrices has an image that is different from the images of all the other matrices in the list?

Short answer

(a)Matrix $L$have the same kernel as matrix $C$.

(b) The matrices that have same image as matrix$C$ are matrices$H$ and$X$ .

(c) The matrix that have the image which is different from the image of all the other matrices is the matrix$L$ .

Step-by-step solution

(a) To find the matrix that have same kernel as matrix C

We have matrix $\mathrm{C}=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]$.

By observing all other matrices, we can say that the matrix that have same kernel as matrix $C$is matrix $L$since its rows are linear combination of rows of matrix $C$.

Thus,$\ker \left(\mathrm{C}\right)=\ker \left(\mathrm{L}\right)$ , where matrix$L$ is given by,

$\mathrm{L}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]$.

(b) To find the matrix that have same image as matrix C

We have matrix$C=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]$ .

By observing all other matrices, we can say that the matrices that have same image as matrix $C$are matrices $H$and $X$since their columns are linear combination of columns of matrix $C$.

Thus, $\mathrm{im}\left(\mathrm{C}\right)=\mathrm{im}\left(\mathrm{H}\right)$and $\mathrm{im}\left(\mathrm{C}\right)=\mathrm{im}\left(\mathrm{X}\right)$, where matrices$H,X$ are given by,

$\mathrm{H}=\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 1\\ 1& 0& 1\end{array}\right],\mathrm{X}=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]$.

(c) To find the image of matrix that is different from image of all other matrices

By observing all other matrices, we can say that the matrix that have the image which is different from the image of all the other matrices is the matrix$L$ where, the matrix$L$ is given by,

$\mathrm{L}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]$

This is because the matrix$T$and matrix $Y$have the same image.

Thus,

$im\left(T\right)=im\left(Y\right)$, where the matrices $T$and$Y$are given by,

$T=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right],Y=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right]$.