If A -4-1-43 0-43-1-3- show that A2-I3-

Here,A2=AA #8658;A2=4 1 430 43 1 34 1 430 43 1 3 #8658;A2=16 3 12 4+0+4 16+4+1212+0 12 3+0+4 12+0+1212 3 9 3+0+3 12+4+9 #8658;A2=100010001 #8756; #160 ...

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Standard XII

Mathematics

Differentiation under Integral Sign

If A =4-1-43 ...

Question

If $A = [\begin{matrix} 4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3 \end{matrix}]$ , show that A² = I₃.

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Solution

$Here, A^{2} = A A \Rightarrow A^{2} = [\begin{matrix} 4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3 \end{matrix}] [\begin{matrix} 4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 16 - 3 - 12 & - 4 + 0 + 4 & - 16 + 4 + 12 \\ 12 + 0 - 12 & - 3 + 0 + 4 & - 12 + 0 + 12 \\ 12 - 3 - 9 & - 3 + 0 + 3 & - 12 + 4 + 9 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] ∴ A^{2} = I_{3}$

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If A=4-1-43 0-43-1-3, show that A2 = I3.

Here,A2=AA⇒A2=4-1-430-43-1-34-1-430-43-1-3⇒A2=16-3-12-4+0+4-16+4+1212+0-12-3+0+4-12+0+1212-3-9-3+0+3-12+4+9⇒A2=100010001∴ A2=I3

If $A = [\begin{matrix} 4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3 \end{matrix}]$ , show that A² = I₃.