Answer to Question #244247 in Linear Algebra for Phamela

Question #244247

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.


1
Expert's answer
2021-10-04T19:39:56-0400

17.

X=(100010001)\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=

[11;11]\begin{bmatrix} 1&-1;&1&-1 \end{bmatrix}


K2 =(1111)\begin{pmatrix} 1 & -1 \\ 1& -1 \end{pmatrix} (1111)\begin{pmatrix} 1 & -1\\ 1& -1 \end{pmatrix}

=(0000)\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=[11;11]\begin{bmatrix} 1&-1&;&1&-1 \end{bmatrix}


That is K(1111)\begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}

K2=(K)(K)


=(1111)\begin{pmatrix} 1 & -1 \\ 1& -1 \end{pmatrix} (1111)\begin{pmatrix} 1 & -1\\ 1 & -1 \end{pmatrix}


(1×1+1×11×1+1×11×1+1×11×1+1×1)\begin{pmatrix} 1×1+-1×1 & 1×-1+-1×-1\\ 1×1+-1×1 & 1×-1+-1×-1 \end{pmatrix}



=(0000)\begin{pmatrix} 0 & 0\\ 0& 0 \end{pmatrix}


=0 (Null matrix)






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