Question

Find X so that for any 3 × 3 real matrix A you get AX = XA = A [Hint : what property is being exhibited by real number p so that for any real w we get wp = pw = w then interpret for matrices.]

Answer 1

Require to find $X$ so that for any $3\times 3$ real matrix $A$ such that $AX=XA=A$

Recollect the following property in real numbers:

If $a$ is any real number, then $a\cdot 1=a=1\cdot a$ and 1 is called the multiplicative identity

Using the above, we have for any real number $w$,

$wp=pw=w$ then $p$ is called the multiplicative identity and $p=1$

Now let us interpret the same for matrices

For any real matrix $A$,

$AX=XA=A$ then the matrix $X$ is called the multiplicative identity and $X$ is identity matrix or unit matrix.

Since $A$ is $3\times 3$ real matrix, so $X$ is also $3\times 3$ matrix

Therefore, $X=\begin{bmatrix} 1 & 0 & 0\\ 0& 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$