Question #147781
Find the sum and product of the matrix of the given eigen values of the matrix
A=2 1 2
1 3 1
2 2 -6
1
Expert's answer
2020-12-01T06:24:59-0500

We have matrix: A=


(212131226)\begin{pmatrix} 2 & 1 & 2 \\ 1 & 3 & 1 \\ 2 & 2 & -6 \end{pmatrix}

Find the eigenvalues of the matrix:

Associate matrix Vlasna vectors values and introduce a variable.

(2λ1213λ1226λ)\begin{pmatrix} 2- \lambda & 1 & 2 \\ 1 & 3-\lambda & 1 \\ 2 & 2 & -6 - \lambda \end{pmatrix}

To do this, find the determinant of the matrix and equate this expression to zero.

(2λ)((3λ)(6λ)21)1(1(6=λ)22)+2(11(3λ)2)=0(2-\lambda)*((3 - \lambda)*(-6-\lambda)-2*1)-1*(1*(-6=\lambda)-2*2)+2*(1*1-(3-\lambda)*2)=0

After transformations, we get:

λ\lambda3+λ+\lambda2 31λ+40=0-31\lambda +40 = 0

After applying the rules, it is clear that the roots of the equation are not integers. Let's calculate the approximate λ\lambda .

λ\lambda1 =6,60=-6,60

λ\lambda2 =4,15=4,15

λ\lambda3 =1,45=1,45

Multiply the matrix A by the matrix of its eigenvalues:

(212131226)\begin{pmatrix} 2 & 1 & 2 \\ 1 & 3 & 1 \\ 2 & 2 & -6 \end{pmatrix} * (6,64,151,45)\begin{pmatrix} -6,6 \\ 4,15 \\ 1,45 \end{pmatrix} == (6,157,313,6)\begin{pmatrix} -6,15 \\ 7,3 \\ -13,6 \end{pmatrix}






Maybe, if I misunderstood something, I add to the report: the sum and product of the matrix A itself:

The sum of the matrix: A + A = 2A

2(212131226)=(4242624412)2\begin{pmatrix} 2 & 1 & 2\\ 1 & 3 & 1\\ 2 & 2 & -6 \end{pmatrix}=\begin{pmatrix} 4 & 2 & 4 \\ 2 & 6 & 2\\ 4 & 4 & -12 \end{pmatrix}


And product of the matrix: AA=A2A * A = A^2


AA=(212131226)(212131226)=(99771216442)A*A=\begin{pmatrix} 2 & 1 & 2 \\ 1 & 3 & 1 \\ 2 & 2 & -6 \end{pmatrix}*\begin{pmatrix} 2 & 1 & 2 \\ 1 & 3 & 1 \\ 2 & 2 & -6 \end{pmatrix}=\begin{pmatrix} 9 & 9 & -7 \\ 7 & 12 & -1 \\ -6 & -4 & 42 \end{pmatrix}


Multiplication is performed for each component of the matrix separately:


a121=a11a11+a12a21+a13a31=22+11+22=9a^2_11=a_11*a_11 +a_12*a_21+a_13*a_31= 2*2+1*1+2*2=9

a122=a11a12+a12a22+a13a32=21+13+22=2+3+4=9a^2_12=a_11*a_12 +a_12*a_22+a_13*a_32=2*1+1*3+2*2=2+3+4=9

a123=a11a13+a12a23+a13a33=22+11+2(6)=7a^2_13=a_11*a_13 +a_12*a_23+a_13*a_33= 2*2+1*1+2*(-6)=-7

a221=a21a11+a22a21+a23a31=12+31+12=7a^2_21=a_21*a_11 +a_22*a_21+a_23*a_31=1*2+3*1+1*2=7

a222=a21a12+a22a22+a23a32=11+33+12=12a^2_22=a_21*a_12 +a_22*a_22+a_23*a_32=1*1+3*3+1*2=12

a223=a21a13+a22a23+a23a33=12+31+1(6)=1a^2_23=a_21*a_13 +a_22*a_23+a_23*a_33=1*2+3*1+1*(-6)=-1

a321=a31a11+a32a21+a33a31=22+21+(6)2=6a^2_31=a_31*a_11 +a_32*a_21+a_33*a_31= 2 *2 +2*1+(-6)*2=-6

a322=a31a12+a32a22+a33a32=21+23+(6)2=4a^2_32=a_31*a_12 +a_32*a_22+a_33*a_32=2*1+2*3+(-6)*2=-4

a323=a31a13+a32a23+a33a33=22+21+(6)(6)=42a^2_33=a_31*a_13 +a_32*a_23+a_33*a_33=2*2+2*1+(-6)*(-6)=42


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