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Question: For the matrix\(A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \end{bmatrix}\), which of th...

For the matrixA=[110121210]A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \end{bmatrix}, which of the following is correct

A

A3+3A2−I=0A^{3} + 3A^{2} - I = 0

B

A3−3A2−I=0A^{3} - 3A^{2} - I = 0

C

A3+2A2−I=0A^{3} + 2A^{2} - I = 0

D

A3−A2+I=0A^{3} - A^{2} + I = 0

Answer

A3−3A2−I=0A^{3} - 3A^{2} - I = 0

Explanation

Solution

A2=A.A=[110121210][110121210]=[231562341]A^{2} = A.A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 3 & 1 \\ 5 & 6 & 2 \\ 3 & 4 & 1 \end{bmatrix}, A3=A.2A=[231562341][110121210]=[793151969124]A^{3} = A.^{2}A = \begin{bmatrix} 2 & 3 & 1 \\ 5 & 6 & 2 \\ 3 & 4 & 1 \end{bmatrix}\begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 7 & 9 & 3 \\ 15 & 19 & 6 \\ 9 & 12 & 4 \end{bmatrix}

7 & 9 & 3 \\ 15 & 19 & 6 \\ 9 & 12 & 4 \end{bmatrix} - \begin{bmatrix} 6 & 9 & 3 \\ 15 & 18 & 6 \\ 9 & 12 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I$$ ⇒ $A^{3} - 3A^{2} - I = 0$