Matrix A is given. Find appropriate identity matrices I m and I n such that I m A = A and AI n = A.
-$$A = \left[ \begin{array} { r r r }
3 & 0 & - 8 \
1 & 8 & - 8 \
- 1 & - 1 & - 2
\end{array} \right]$$
A) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \ 0 & 1 \end{array} \right]$
B) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$
C) $\mathrm { I } _ { \mathrm { m } } = \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$
D) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$

Question

Matrix A is given. Find appropriate identity matrices I m and I n such that I m A = A and AI n = A.
-$$A = \left[ \begin{array} { r r r } 
3 & 0 & - 8 \
1 & 8 & - 8 \
- 1 & - 1 & - 2
\end{array} \right]$$
A) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \ 0 & 1 \end{array} \right]$
B) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$
C) $\mathrm { I } _ { \mathrm { m } } = \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$
D) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$

Quizplus · Accepted Answer

The identity matrix that satisfies both $IM = M$ and $MI = M$ for any matrix $M$ must match the dimensions of $M$. Since $A$ is a $3 \times 3$ matrix, the appropriate identity matrices for both pre-multiplication and post-multiplication must also be $3 \times 3$ to maintain the dimensions of $A$.

Matrix a Is Given A=[30−818−8−1−1−2]A = \left[ \begin{array} { r r r } 3 & 0 & - 8 \\1 & 8 & - 8 \\- 1 & - 1 & - 2\end{array} \right]A=​31−1​08−1​−8−8−2​​

Matrix a Is Given $A = \left[ \begin{array} { r r r } 3 & 0 & - 8 \\1 & 8 & - 8 \\- 1 & - 1 & - 2\end{array} \right]$