Question

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

Answer 1

★ Correct Question :-

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

★ Concept :-

Here the concept of Matrices and Matrice function has been used. We see that we are given a matrix. We are also given a function defined by f(x). We have to find the value of f(A). So firstly we can apply the value of A that isnthe matrix in the function. Then we can use matrix operations and thus find the value. Finally we can solve matrix and then find the correct option.

Let's do it !!

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★ Solution :-

• Correct matrix ::

$\bf{\green{A = \left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]}}$

• f(x) = 2x² - 3x

We see that the function is defined by f(x). So here if we replace the value of x by A, we will get

• f(A) = 2A² - 3A

Now we need to simply apply the matrix here. Then,

$\tt{f(A)\;=\;2\left(\left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]\right)^{2}\;-\;3\left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]}$

Here A² means that matrix A is multiplied with itself. Then,

$\tt{f(A)\;=\;2\left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]\left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]\;-\;3\left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]}$

$\tt{f(A)\;=\;2\left[\begin{array}{c c c} 4\:+\:0 & -2\:+\:3 \\ \\ 0\:+\:0 & 0\:+\:9 \\ \\ \end{array}\right]\;-\;3\left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]}$

$\tt{f(A)\;=\;2\left[\begin{array}{c c c} 4 & 1 \\ \\ 0 & 9 \\ \\ \end{array}\right]\;-\;3\left[\begin{array}{c c c} -2 & 1 \\ \\ 0 & 3 \\ \\ \end{array}\right]}$

Now we know that if any number is multiplied to a matrix then it is multiplied to every element of the matrix. So,

$\tt{f(A)\;=\;\left[\begin{array}{c c c} 2(4) & 2(1) \\ \\ 2(0) & 2(9) \\ \\ \end{array}\right]\;-\;\left[\begin{array}{c c c} 3(-2) & 3(1) \\ \\ 3(0) & 3(3) \\ \\ \end{array}\right]}$

$\tt{f(A)\;=\;\left[\begin{array}{c c c} 8 & 2 \\ \\ 0 & 18 \\ \\ \end{array}\right]\;-\;\left[\begin{array}{c c c} -6 & 3 \\ \\ 0 & 9 \\ \\ \end{array}\right]}$

We already know that only adjoining pairs of two matrices get added or subtracted. This means that, the element of a matrix with a row and column can be added or subtracted to element with same row and same column in other matrix. So,

$\tt{f(A)\;=\;\left[\begin{array}{c c c} [8\:-\:(-6)]\;\;&\;\;[2\:-\:3]\\\\0\;\;&\;\;[18\:-\:9]\\\\\end{array}\right]}$

$\tt{f(A)\;=\;\left[\begin{array}{c c c}[8\:+\:6]\;\;&\;\;[2\:-\:3]\\\\0\;\;&\;\;[18\:-\:9]\\\\ \end{array}\right]}$

$\tt{f(A)\;=\;\left[\begin{array}{c c c} [14] & [-1] \\ \\ 0 & [9] \\ \\ \end{array}\right]}$

$\bf{\red{f(A)\;=\;\left[\begin{array}{c c c} 14 & -1 \\ \\ 0 & 9 \\ \\ \end{array}\right]}}$

This is the required answer.

So, option C. is correct.

$\;\underline{\boxed{\tt{Required\;\:value\;=\;\bf{\purple{\left[\begin{array}{c c c} 14 & -1 \\ \\ 0 & 9 \\ \\ \end{array}\right]}}}}}$