If A=≤ft[{array}{cc}1 & -2 \\ 4 & 5{array}] ; f(t)=t²-3 t+7 then f(A)+≤ft[{array}{cc}3 & 6 \\ -12 & -9{array}]=

Correct Answer is : [0000]≤ft[{array}{ll}0 & 0 \\ 0 & 0{array}][0000]

≤ft[{array}{ll}0 & 1 \\ 1 & 0{array}]

≤ft[{array}{ll}1 & 1 \\ 0 & 0{array}]

≤ft[{array}{ll}0 & 0 \\ 0 & 0{array}]

≤ft[{array}{ll}1 & 0 \\ 0 & 1{array}]

Correct Answer is : [0000]≤ft[{array}{ll}0 & 0 \\ 0 & 0{array}][0000]

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COMEDK 2025 Morning Solved Paper

Question : 142 of 180

Marks: +1, -0

A=\left[\begin{array}{cc}1 & -2 \\ 4 & 5\end{array}\right] ; f(t)=t^2-3 t+7

f(A)+\left[\begin{array}{cc}3 & 6 \\ -12 & -9\end{array}\right]=

Solution:

Concept:

Compute the matrix polynomial

f(A) = A^2 - 3A + 7I

and then add the given matrix.

Explanation:

First, find

A^2

A = \begin{pmatrix}1 & -2 \\ 4 & 5\end{pmatrix}

A^2 = \begin{pmatrix}1 & -2 \\ 4 & 5\end{pmatrix} \begin{pmatrix}1 & -2 \\ 4 & 5\end{pmatrix} = \begin{pmatrix}1\cdot1 + (-2)\cdot4 & 1\cdot(-2) + (-2)\cdot5 \\ 4\cdot1 + 5\cdot4 & 4\cdot(-2) + 5\cdot5\end{pmatrix} = \begin{pmatrix}-7 & -12 \\ 24 & 17\end{pmatrix}

Next, compute

-3A

and

7I

-3A = -3\begin{pmatrix}1 & -2 \\ 4 & 5\end{pmatrix} = \begin{pmatrix}-3 & 6 \\ -12 & -15\end{pmatrix}

7I = \begin{pmatrix}7 & 0 \\ 0 & 7\end{pmatrix}

Now,

f(A) = A^2 - 3A + 7I = \begin{pmatrix}-7 & -12 \\ 24 & 17\end{pmatrix} + \begin{pmatrix}-3 & 6 \\ -12 & -15\end{pmatrix} + \begin{pmatrix}7 & 0 \\ 0 & 7\end{pmatrix} = \begin{pmatrix}-3 & -6 \\ 12 & 9\end{pmatrix}

Add the given matrix:

f(A) + \begin{pmatrix}3 & 6 \\ -12 & -9\end{pmatrix} = \begin{pmatrix}-3+3 & -6+6 \\ 12-12 & 9-9\end{pmatrix} = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}

Answer:

\begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}

, which corresponds to option C.

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