To solve for the identity matrices [tex]\( I_m \)[/tex] and [tex]\( I_n \)[/tex] that satisfy [tex]\( I_m A = A \)[/tex] and [tex]\( A I_n = A \)[/tex] respectively, let's follow these steps:
1. Understand the structure of the given matrix [tex]\( A \)[/tex]:
[tex]\[
A = \left[\begin{array}{rrrr}
2 & -1 & 0 & 3 \\
-2 & 6 & -2 & 7 \\
5 & -5 & 7 & 11
\end{array}\right]
\][/tex]
2. Determine the dimensions of [tex]\( A \)[/tex]:
- [tex]\( A \)[/tex] has 3 rows and 4 columns, so [tex]\( A \)[/tex] is a [tex]\( 3 \times 4 \)[/tex] matrix.
3. Identify the appropriate identity matrix [tex]\( I_m \)[/tex] for [tex]\( I_m A = A \)[/tex]:
- Since [tex]\( I_m \)[/tex] must be a [tex]\( 3 \times 3 \)[/tex] matrix to match the number of rows of [tex]\( A \)[/tex], [tex]\( I_m \)[/tex] should be the [tex]\( 3 \times 3 \)[/tex] identity matrix.
- The identity matrix [tex]\( I_m \)[/tex] is:
[tex]\[
I_m = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
\][/tex]
4. Identify the appropriate identity matrix [tex]\( I_n \)[/tex] for [tex]\( A I_n = A \)[/tex]:
- Since [tex]\( I_n \)[/tex] must be a [tex]\( 4 \times 4 \)[/tex] matrix to match the number of columns of [tex]\( A \)[/tex], [tex]\( I_n \)[/tex] should be the [tex]\( 4 \times 4 \)[/tex] identity matrix.
- The identity matrix [tex]\( I_n \)[/tex] is:
[tex]\[
I_n = \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\][/tex]
5. Verify the results:
- Multiplying [tex]\( I_m \)[/tex] with [tex]\( A \)[/tex] should leave [tex]\( A \)[/tex] unchanged.
- Multiplying [tex]\( A \)[/tex] with [tex]\( I_n \)[/tex] should also leave [tex]\( A \)[/tex] unchanged.
So the identity matrices are:
[tex]\[
I_m = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
\][/tex]
[tex]\[
I_n = \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\][/tex]
