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### Answers (1)

option (d) none of these

Given: $A=\begin{bmatrix} 1 &0 &1 \\ 0& 0& 1\\ a&b &2 \end{bmatrix}$

Hint:  identity matrix is that whose diagonal elements is 1 and rest and zero

Solution:

\begin{aligned} &A^{2}=\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2 \end{array}\right] \\ &=\left[\begin{array}{ccc} 1+a & b & 3 \\ a & b & 2 \\ 3 a & 2 b & a+b+4 \end{array}\right] \\ &\text { Now, a I }+b A+2 A^{2} \\ &=\left[\begin{array}{lll} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{array}\right]+\left[\begin{array}{ccc} b & 0 & b \\ 0 & 0 & b \\ a b & b^{2} & 2 b \end{array}\right]+\left[\begin{array}{ccc} 2+2 a & 2 b & 6 \\ 2 a & 2 b & 4 \\ 6 a & 4 b & 2 a+2 b+8 \end{array}\right] \\ &=\left[\begin{array}{ccc} 3 a+b+2 & 2 b & b+6 \\ 2 a & a+2 b & b+4 \\ a b+6 a & b^{2}+4 b & 3 a+4 b+8 \end{array}\right] \end{aligned}

Now, value of a & b is not given so it is not solved further

Hence, option (d) is correct

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