Let ‘M’ be the set of all possible 2×2 matrices A of integer entries such that T AAT =I where I=1 00 1 then (where det A denotes determinant value of matrix A)

AA r = 1 Let A = a b c d a 2 + b 2 = 1 c 2 + d 2 = 1 ac + bd = 0 ⇒ a = ± 1 , b = 0 , c = 0 , d = ± 1 a = 0 , b = ± 1 , c = ± 1 , d = 0 Total 8 matrices | A − I | = 0 ⇒ ad − bc + 1 = a + d ⇒ a = d = 1 a = 1 d = − 1 a = − 1 d = 1 ⇒ regd. no. = 8 − 3 = 5

Number of matrices in set M such that det.(A – I) ≠ 0 is 2

Number of matrices in set M such that det. (A – I) ≠ 0 is 3

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Let ‘M’ be the set of all possible $2 \times 2$ matrices A of integer entries such that T AA^T =I where $I = (\begin{array}{ll} 1 0 \\ 0 1 \end{array})$ then (where det A denotes determinant value of matrix A)

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Number of matrices in set M is 8

Number of matrices in set M such that det.(A – I) $\neq$ 0 is 2

Number of matrices in set M is 4

Number of matrices in set M such that det. (A – I) $\neq$ 0 is 3

answer is A, C.

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Detailed Solution

${AA}^{r} = 1$ Let $A = [\begin{array}{ll} a b \\ c d \end{array}]$
$\begin{array}{l} a^{2} + b^{2} = 1 \\ c^{2} + d^{2} = 1 \\ ac + bd = 0 \\ \Rightarrow a = \pm 1, b = 0, c = 0, d = \pm 1 \\ a = 0, b = \pm 1, c = \pm 1, d = 0 \end{array}$
Total 8 matrices
$\begin{array}{l} | A - I | = 0 \\ \Rightarrow ad - bc + 1 = a + d \\ \Rightarrow a = d = 1 \\ a = 1 d = - 1 \\ a = - 1 d = 1 \\ \Rightarrow regd. no. = 8 - 3 = 5 \end{array}$

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