Answer to Question #244247 in Linear Algebra for Phamela

Question

7.) 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.] 1 8.) Consider K =  1 −1 1 −1  then we get K2 = 0 Does this hold for real numbers? Motivate.

Answer 1

17.

X=$\begin{pmatrix} 1&0&0 \\ 0&1&0\\ 0&0&1 \end{pmatrix}$

Its equivalent to identify property of multiplication in real numbers where:

wp=pw=w given p= 1

Given K=

$\begin{bmatrix} 1&-1;&1&-1 \end{bmatrix}$

K² =$\begin{pmatrix} 1 & -1 \\ 1& -1 \end{pmatrix}$ $\begin{pmatrix} 1 & -1\\ 1& -1 \end{pmatrix}$

=$\begin{pmatrix} 0 & 0 \\ 0& 0 \end{pmatrix}$

The above is not true for real numbers.

Only a square of "0" will give a zero

0$×$0=0.

18.

-If ab=0, either a=0 or b=0

-Products of two non-zero numbers is always non-zero

But products of two non-zero matrices can be zero matrix

Using K given above

K=$\begin{bmatrix} 1&-1&;&1&-1 \end{bmatrix}$

That is K$\begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}$

K²=(K)(K)

=$\begin{pmatrix} 1 & -1 \\ 1& -1 \end{pmatrix}$ $\begin{pmatrix} 1 & -1\\ 1 & -1 \end{pmatrix}$

$\begin{pmatrix} 1×1+-1×1 & 1×-1+-1×-1\\ 1×1+-1×1 & 1×-1+-1×-1 \end{pmatrix}$

=$\begin{pmatrix} 0 & 0\\ 0& 0 \end{pmatrix}$

=0 (Null matrix)