Question

$\sf If\; A = \left [ \begin{array}{c c c} - & 1 \\ \\ 0 & 9 \\ \\ \end {array} \right]\sf and\; f(x) = 2x^2 -3x\; then\; f(A) = a) \left [ \begin{array}{c c c} 14 & 1 \\ \\ 0 & -9 \\ \\ \end {array} \right] b) \left [ \begin{array}{c c c} -14 & 1 \\ \\ 0 & 9 \\ \\ \end {array} \right] c) \left [ \begin{array}{c c c} 14 & 1 \\ \\ 0 & 9 \\ \\ \end {array} \right] d) \left [ \begin{array}{c c c} -14 & -1 \\ \\ 0 & -9 \\ \\ \end {array} \right]$​

Answer 1

Correct Question :-

$\begin{gathered}\sf If\;A = \left [ \begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \end {array} \right]\sf and\; f(x) = 2 {x}^{2} -3x\;then\;f(A) = \\\\\\ \sf a) \left [ \begin{array}{c c c} 14 & 1 \\ \\ 0 & -9 \end {array} \right] \\\\\\ \sf b) \left [ \begin{array}{c c c} -14 & 1 \\ \\ 0 & 9 \end {array} \right] \\\\\\ \sf c) \left [ \begin{array}{c c c} 14 & -1 \\ \\ 0 & 9 \end {array} \right] \\\\\\ \sf d) \left [ \begin{array}{c c c} -14 & -1 \\ \\ 0 & -9 \end {array} \right]\end{gathered}$

Solution :-

Given that

$\rm :\longmapsto\:A = \: \begin{bmatrix} - 2 & 1\\ 0 & 3\end{bmatrix}$

Consider,

$\rm :\longmapsto\: \: {A}^{2}$

$\: \sf \: \: = \: \: A \times A$

$\: \sf \: \: = \: \: \: \begin{bmatrix} - 2 & 1\\ 0 & 3\end{bmatrix} \times \: \begin{bmatrix} - 2 & 1\\ 0 & 3\end{bmatrix}$

$\: \sf \: \: = \: \: \: \begin{bmatrix} - 2 \times ( - 2) + 1 \times 0 & - 2 \times 1 + 1 \times 3\\ 0 \times ( - 2) + 3 \times 0 & 0 \times 1 + 3 \times 3\end{bmatrix}$

$\: \sf \: \: = \: \: \: \begin{bmatrix} 4 & 1\\ 0 & 9\end{bmatrix}$

Now,

We have to given that

$\rm :\longmapsto\:f(x) = {2x}^{2} - 3x$

Thus,

$\rm :\longmapsto\:f(A) = {2A}^{2} - 3A$

$\: \sf \: \: = \: \: \: 2\begin{bmatrix} 4 & 1\\ 0 & 9\end{bmatrix} - 3\: \begin{bmatrix} - 2 & 1\\ 0 & 3\end{bmatrix}$

$\: \sf \: \: = \: \: \: \begin{bmatrix} 8 & 2\\ 0 & 18\end{bmatrix} - \: \begin{bmatrix} - 6 & 3\\ 0 & 9\end{bmatrix}$

$\: \sf \: \: = \: \: \: \begin{bmatrix} 8 + 6 & 2 - 3\\ 0 - 0 & 18 - 9\end{bmatrix}$

$\: \sf \: \: = \: \: \: \begin{bmatrix} 14 & - 1\\ 0 & 9\end{bmatrix}$

$\rm :\implies\:Option \: (c) \: is \: correct.$

Additional Information :-

1. Matrix multiplication is not Commutative always, i.e. AB may or may not be equal to BA.

2. Matrix multiplication is associative, i.e A(BC) = (AB)C.

3. Matrix multiplication is distributive, i.e A(B + C) = AB + AC.

4. There exist a matrix I such that AI = IA = A.