If A-1 0 0 0-1 0 00-1- find A3-

Here,A2=AA #8658;A2= 1000 1000 1 1000 1000 1 #8658;A2=1+0+00+0+00+0+00+0+00+1+00+0+00+0+00+0+00+0+1 #8658;A2=100010001Now, #160;A3=A2A #8658;A3=100010 ...

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Standard XII

Mathematics

Properties of Determinants

If A=-1 0 0...

Question

If $A = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$ , find A³.

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Solution

$Here, A^{2} = A A \Rightarrow A^{2} = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 1 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 1 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 1 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] Now, A^{3} = A^{2} A \Rightarrow A^{3} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] \Rightarrow A^{3} = [\begin{matrix} - 1 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 - 1 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 0 + 0 & 0 + 0 - 1 \end{matrix}] \Rightarrow A^{3} = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] = A$

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Similar questions

Q. If

[\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]

, find x, y and z.

Q. If

[x]

stands for the greatest integer function, then

[\frac{1}{2} + \frac{1}{1000}] + [\frac{1}{2} + \frac{2}{1000}] + \dots . + [\frac{1}{2} + \frac{999}{1000}] =

Assertion (A) :

[\frac{1}{2}] + [\frac{1}{2} + \frac{1}{1000}] + [\frac{1}{2} + \frac{2}{1000}] + [\frac{1}{2} + \frac{3}{1000}] + . . . + [\frac{1}{2} + \frac{999}{1000}] = 500

(where [.] denotes G.I.F.)

Reason (R) :

[\frac{1}{2} + \frac{r}{1000}] = {\begin{matrix} 0; & if 0 \leq r < 500 1; & if 500 \leq r \leq 999 \end{matrix}

Find the value of $^{1000} C_{500} +^{1000} C_{501}$

Q. If

A = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]

, find A².

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If A=-1 0 0 0-1 0 0 0-1, find A3.

Here,A2=AA⇒A2=-1000-1000-1-1000-1000-1⇒A2=1+0+00+0+00+0+00+0+00+1+00+0+00+0+00+0+00+0+1⇒A2=100010001Now, A3=A2A⇒A3=100010001-1000-1000-1⇒A3=-1+0+00+0+00+0+00+0+00-1+00+0+00+0+00+0+00+0-1⇒A3=-1000-1000-1=A

If $A = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$ , find A³.