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Standard X

Mathematics

Multiplication of Matrices

Let a matrix ...

Question

Let a matrix $A = [\begin{matrix} - 1 & - 5 0 & 1 \end{matrix}],$ then which of the following statement(s) is(are) correct for natural number $n \in N$
(where $I$ is unit matrix of order $2.$ )

A^{2 n} = A

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A^{2 n + 1} = A

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A^{2 n} = I

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A^{2 n + 1} = I

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Solution

The correct option is C $A^{2 n} = I$
Given : $A = [\begin{matrix} - 1 & - 5 0 & 1 \end{matrix}]$
$\Rightarrow A^{2} = [\begin{matrix} - 1 & - 5 0 & 1 \end{matrix}] [\begin{matrix} - 1 & - 5 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]$
$\Rightarrow A^{2} = I$
So, $A^{4} = I^{2} = I$
Similarly, $A^{2 n} = I, n \in N$
and $A^{3} = A^{2} A = I A = A$
similarly, $A^{2 n + 1} = A, n \in N$

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Let a matrix A=[−1−501], then which of the following statement(s) is(are) correct for natural number n∈N (where I is unit matrix of order 2.)

The correct option is C A2n=IGiven : A=[−1−501] ⇒A2=[−1−501][−1−501]=[1001] ⇒A2=I So, A4=I2=I Similarly, A2n=I,n∈N and A3=A2A=IA=A similarly, A2n+1=A,n∈N

Let a matrix $A = [\begin{matrix} - 1 & - 5 0 & 1 \end{matrix}],$ then which of the following statement(s) is(are) correct for natural number $n \in N$
(where $I$ is unit matrix of order $2.$ )