Răspuns :
a=2 => A(a)=( 2 4)
( 3 -2)
A^2= A*A= (2 4) *(2 4)= (16 0)
(3 -2) (3 -2) (0 16)
det(A(2))= 2 4 = -4 -12 = -16
3 -2
det (A(2))* I2= -16* (1 0) = ( -16 0) => A^2 = det(A(a))*i2
(0 1) (0 -16)
( 3 -2)
A^2= A*A= (2 4) *(2 4)= (16 0)
(3 -2) (3 -2) (0 16)
det(A(2))= 2 4 = -4 -12 = -16
3 -2
det (A(2))* I2= -16* (1 0) = ( -16 0) => A^2 = det(A(a))*i2
(0 1) (0 -16)
[tex]\displaystyle \mathtt{A(a)= \left(\begin{array}{ccc}\mathtt a&\mathtt4\\\mathtt3&\mathtt{-a}\end{array}\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A^2=-det(A) \cdot I_2}\\ \\ \mathtt{a=2 \Rightarrow A(2)= \left(\begin{array}{ccc}\mathtt 2&\mathtt4\\\mathtt3&\mathtt{-2}\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{A^2= \left(\begin{array}{ccc}\mathtt 2&\mathtt4\\\mathtt3&\mathtt{-2}\end{array}\right) \cdot \left(\begin{array}{ccc}\mathtt 2&\mathtt4\\\mathtt3&\mathtt{-2}\end{array}\right) }= \\ \\ =\mathtt{\left(\begin{array}{ccc}\mathtt {2 \cdot 2+4 \cdot 3}&\mathtt{2 \cdot 4+4 \cdot (-2)}\\\mathtt{3 \cdot 2+(-2) \cdot 3}&\mathtt{3 \cdot 4+(-2) \cdot (-2)}\end{array}\right)= \left(\begin{array}{ccc}\mathtt {4+12}&\mathtt{8-8}\\\mathtt{6-6}&\mathtt{12+4}\end{array}\right)=}[/tex]
[tex]\displaystyke \mathtt{= \left(\begin{array}{ccc}\mathtt {16}&\mathtt0\\\mathtt0&\mathtt{16}\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{det(A)=\left|\begin{array}{ccc}\mathtt {2}&\mathtt4\\\mathtt3&\mathtt{-2}\end{array}\right|=2 \cdot (-2)-4 \cdot 3=-4-12=-16}\\ \\ \mathtt{det(A)=-16 \Rightarrow -det(A)=16}}[/tex]
[tex]\displaystyle \mathtt{-det(A) \cdot I_2=16 \cdot \left(\begin{array}{ccc}\mathtt {1}&\mathtt0\\\mathtt0&\mathtt{1}\end{array}\right)=\left(\begin{array}{ccc}\mathtt {16 \cdot 1}&\mathtt16 \cdot0\\\mathtt16\cdot0&\mathtt{16\cdot1}\end{array}\right)=\left(\begin{array}{ccc}\mathtt {16}&\mathtt0\\\mathtt0&\mathtt{16}\end{array}\right) }\\ \\ \mathtt{\Rightarrow A^2=-det(A) \cdot I_2}[/tex]
[tex]\displaystyle \mathtt{A^2= \left(\begin{array}{ccc}\mathtt 2&\mathtt4\\\mathtt3&\mathtt{-2}\end{array}\right) \cdot \left(\begin{array}{ccc}\mathtt 2&\mathtt4\\\mathtt3&\mathtt{-2}\end{array}\right) }= \\ \\ =\mathtt{\left(\begin{array}{ccc}\mathtt {2 \cdot 2+4 \cdot 3}&\mathtt{2 \cdot 4+4 \cdot (-2)}\\\mathtt{3 \cdot 2+(-2) \cdot 3}&\mathtt{3 \cdot 4+(-2) \cdot (-2)}\end{array}\right)= \left(\begin{array}{ccc}\mathtt {4+12}&\mathtt{8-8}\\\mathtt{6-6}&\mathtt{12+4}\end{array}\right)=}[/tex]
[tex]\displaystyke \mathtt{= \left(\begin{array}{ccc}\mathtt {16}&\mathtt0\\\mathtt0&\mathtt{16}\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{det(A)=\left|\begin{array}{ccc}\mathtt {2}&\mathtt4\\\mathtt3&\mathtt{-2}\end{array}\right|=2 \cdot (-2)-4 \cdot 3=-4-12=-16}\\ \\ \mathtt{det(A)=-16 \Rightarrow -det(A)=16}}[/tex]
[tex]\displaystyle \mathtt{-det(A) \cdot I_2=16 \cdot \left(\begin{array}{ccc}\mathtt {1}&\mathtt0\\\mathtt0&\mathtt{1}\end{array}\right)=\left(\begin{array}{ccc}\mathtt {16 \cdot 1}&\mathtt16 \cdot0\\\mathtt16\cdot0&\mathtt{16\cdot1}\end{array}\right)=\left(\begin{array}{ccc}\mathtt {16}&\mathtt0\\\mathtt0&\mathtt{16}\end{array}\right) }\\ \\ \mathtt{\Rightarrow A^2=-det(A) \cdot I_2}[/tex]
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