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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{bm...

If a function f(x) =x2−2xx^{2}-2x, find f(A), where A=[012 450 023]A=\begin{bmatrix}0&1&2\\\ 4&5&0\\\ 0&2&3\end{bmatrix}.

Explanation

Solution

Hint: In this question it is given that if f(x)=x2−2xf\left( x\right) =x^{2}-2x, we have to find f(A),
where A=[012 450 023]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×A2\times A and A2A^{2}=A×AA\times A, so for this we need to know the multiplication method for two matrices.
If A=[aij]m×nA=\left[ a_{ij}\right]_{m\times n} be matrix of order m×nm\times n and B=[bij]n×pB=\left[ b_{ij}\right]_{n\times p} of order n×pn\times p , then,
A×B=[cij]m×pA\times B=\left[ c_{ij}\right]_{m\times p} , where cij=ai1b1j+ai2b2j+⋯+ainbnjc_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots +a_{in}b_{nj}.
Also here aija_{ij} defines the element of ithi^{th} and jthj^{th} columns.
Complete step-by-step solution:
Given,
A=[012 450 023]A=\begin{bmatrix}0&1&2\\\ 4&5&0\\\ 0&2&3\end{bmatrix}
Therefore,
A2A^{2}
=A×A=A\times A
=[012 450 023]×[012 450 023]=\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}
=[0×0+1×4+2×00×1+1×5+2×20×2+1×0+2×3 4×0+5×4+0×04×1+5×5+0×24×2+5×0+0×3 0×0+2×4+3×00×1+2×5+3×20×2+2×0+3×3]=\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}
=[496 20298 8169]=\begin{bmatrix}4&9&6\\\ 20&29&8\\\ 8&16&9\end{bmatrix}
Now,
2A=2[012 450 023]=2\begin{bmatrix}0&1&2\\\ 4&5&0\\\ 0&2&3\end{bmatrix}
=[2×02×12×2 2×42×52×0 2×02×22×3]=\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}
=[024 8100 046]=\begin{bmatrix}0&2&4\\\ 8&10&0\\\ 0&4&6\end{bmatrix}
Now,
f(A)=A2−2Af\left(A\right) =A^{2}-2A
=[496 20298 8169]−[024 8100 046]=\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}
=[4−09−26−4 20−829−108−0 8−016−49−6]=\begin{bmatrix}4-0&9-2&6-4\\\ 20-8&29-10&8-0\\\ 8-0&16-4&9-6\end{bmatrix}
=[472 12198 8123]=\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.