Question

If a function f(x) =$x^{2}-2x$, find f(A), where $$A=\begin{bmatrix}0&1&2\\ 4&5&0\\ 0&2&3\end{bmatrix}$$.

Answer 1

Hint: In this question it is given that if $f\left( x\right) =x^{2}-2x$, we have to find f(A),
where $$A=\begin{bmatrix}0&1&2\\ 4&5&0\\ 0&2&3\end{bmatrix}$$.
So to find the solution we first need to find $2\times A$ and $A^{2}$=$A\times A$, so for this we need to know the multiplication method for two matrices.
If $$A=\left[ a_{ij}\right]_{m\times n} $$ be matrix of order $m\times n$ and $$B=\left[ b_{ij}\right]_{n\times p} $$ of order $n\times p$ , then,
$$A\times B=\left[ c_{ij}\right]_{m\times p} $$, where $$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots +a_{in}b_{nj}$$.
Also here $a_{ij}$ defines the element of $i^{th}$ and $j^{th}$ columns.
Complete step-by-step solution:
Given,
$$A=\begin{bmatrix}0&1&2\\ 4&5&0\\ 0&2&3\end{bmatrix}$$
Therefore,
$A^{2}$
$=A\times A$
$$=\begin{bmatrix}0&1&2\\ 4&5&0\\ 0&2&3\end{bmatrix} \times \begin{bmatrix}0&1&2\\ 4&5&0\\ 0&2&3\end{bmatrix}$$
$$=\begin{bmatrix}0\times 0+1\times 4+2\times 0&0\times 1+1\times 5+2\times 2&0\times 2+1\times 0+2\times 3\\ 4\times 0+5\times 4+0\times 0&4\times 1+5\times 5+0\times 2&4\times 2+5\times 0+0\times 3\\ 0\times 0+2\times 4+3\times 0&0\times 1+2\times 5+3\times 2&0\times 2+2\times 0+3\times 3\end{bmatrix}$$
$$=\begin{bmatrix}4&9&6\\ 20&29&8\\ 8&16&9\end{bmatrix}$$
Now,
2A$$=2\begin{bmatrix}0&1&2\\ 4&5&0\\ 0&2&3\end{bmatrix}$$
 $$=\begin{bmatrix}2\times 0&2\times 1&2\times 2\\ 2\times 4&2\times 5&2\times 0\\ 2\times 0&2\times 2&2\times 3\end{bmatrix}$$
$$=\begin{bmatrix}0&2&4\\ 8&10&0\\ 0&4&6\end{bmatrix}$$
Now,
 $f\left(A\right) =A^{2}-2A$
$$=\begin{bmatrix}4&9&6\\ 20&29&8\\ 8&16&9\end{bmatrix} -\begin{bmatrix}0&2&4\\ 8&10&0\\ 0&4&6\end{bmatrix}$$
$$=\begin{bmatrix}4-0&9-2&6-4\\ 20-8&29-10&8-0\\ 8-0&16-4&9-6\end{bmatrix}$$
$$=\begin{bmatrix}4&7&2\\ 12&19&8\\ 8&12&3\end{bmatrix}$$
Which is our required solution.
Note: while performing addition and subtraction in two matrices, the operation takes place in its corresponding elements and when you multiply a constant term with a matrix this will multiply with each and every element of a matrix.