Let $$A=$$ $$2$$ $$3$$−$$1$$ $$2$$ and $$f(x)=x2$$−$$4x+7$$ Show that $$f(A)=O$$ Use this result to find $$A5$$

Question

Let $$A=$$  $$2$$    $$3$$−$$1$$    $$2$$ and $$f(x)=x2$$−$$4x+7$$  Show that $$f(A)=O$$ Use this result to find $$A5$$

Infinity Learn · Accepted Answer

we have,$$f(x)=x2$$−$$4x+7$$.Therefore, $$f(A)=A2$$−$$4A+7I2A2=$$  $$2$$    $$3$$−$$1$$    $$2$$  $$2$$    $$3$$−$$1$$    $$2=$$  $$4-3$$   $$6+6$$−$$2$$−$$2$$ $$-3+4=$$  $$1$$   $$12$$−$$4$$   $$1$$  −$$4A=-8$$   $$-124$$     $$-8$$  $$and7I2=7$$  $$00$$   $$7$$∴$$f(A)=A2$$−$$4A+7I2=$$  $$1$$   $$12$$−$$4$$   $$1$$  $$+-8$$   $$-124$$     $$-8$$  $$+7$$  $$00$$   $$7=$$   $$1-8+7$$      $$12-12+0$$−$$4+4+0$$     $$1-8+7$$  $$=0$$  $$00$$   $$0=Of(A)=O$$⇒$$A2$$−$$4A+7I2=O$$⇒$$A2=4A$$−$$7I2$$⇒$$A3=A2$$.$$A=(4A$$−$$7I2)A=4A2$$−$$7I2A$$⇒$$A3=44A-7I2-7A$$⇒$$A3=9A-28I2$$⇒$$A4=A3A=9A-28I2A$$⇒$$A4=9A2-28A=94A-7I2-28A$$⇒$$A4=36A-63I2-28A=8A-63I2$$⇒$$A5=A4$$·$$A=8A-63I2A=8A2-63I2A$$⇒$$A5=84A-7I2-63A=-31A-56I2$$⇒$$A5=-3123-12-561001$$⇒$$A5=-62-9331-62-560056$$⇒$$A5=-118-9331-118$$

Let $A = [\begin{array}{l} 2 3 \\ - 1 2 \end{array}]$ and $f (x) = x^{2} - 4 x + 7$ Show that $f (A) = O$ Use this result to find $A^{5}$

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Let A= 2 3−1 2 and f(x)=x2−4x+7 Show that f(A)=O Use this result to find A5

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Let $A = [\begin{array}{l} 2 3 \\ - 1 2 \end{array}]$ and $f (x) = x^{2} - 4 x + 7$ Show that $f (A) = O$ Use this result to find $A^{5}$