Let $A = \begin{bmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 1 & 1 \end{bmatrix}$ and $B = A^{20}$. Then the sum of the elements of the first column of $B$ is

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Aicharya · Accepted Answer

A &#160; = 0 - 1 1 0 A 2 = &#160; 0 - 1 1 0 0 - 1 1 0 = &#160; - 1 0 0 - 1 A 3 = - 1 0 0 - 1 0 - 1 1 0 = 0 1 - 1 0 A 4 = 0 1 - 1 0 0 - 1 1 0 = 1 0 0 1 Option 1 A 3 + I = &#160; A &#160; A 3 - I &#160; &#160; 0 1 &#160; - 1 0 &#160; &#160; - 1 - 1 &#160; &#160; 1 - 1 = &#160; 1 - 1 &#160; 1 1 1 - 1 &#160; 1 1 = 1 - 1 &#160; 1 1 .....Correct Option 2 A 4 - I = A 2 + I 0 0 0 0 = &#160; 0 0 0 0 ....Correct Option 3 0 0 0 0 = &#160; 0 - 1 1 0 &#215; &#160; - 2 0 0 - 2 &#160; = &#160; 0 2 - 2 0 .....Incorrect Option 4 A 3 - I = A ( A - I ) - 1 1 - 1 - 1 = &#160; 0 - 1 1 0 &#160; &#215; &#160; - 1 - 1 1 - 1 &#160; = &#160; - 1 1 - 1 - 1 ......Correct.

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$ and $B = A^{20}$. Then the sum…

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