(B) matrix $A$ is orthogonal if $A^T A = I$. Taking the determinant on both sides, we get $|A^T A| = |I|$. Since $|A^T| = |A|$, we have $|A|^2 = 1$, w

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(B) matrix $A$ is orthogonal if $A^T A = I$. Taking the determinant on both sides, we get $|A^T A| = |I|$. Since $|A^T| = |A|$, we have $|A|^2 = 1$, which implies $|A| = \pm 1$. Since $|A| \neq 0$, every orthogonal matrix is non-singular. Thus, $Q \subseteq P$. Since there exist non-singular matrices that are not orthogonal (e.g.,any diagonal matrix with entries other than $1$ or $-1$), $Q$ is a proper subset of $P$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Types of matrices, Algebra of matrices

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Let $P$ be the set of all non-singular matrices of order $3$ over $\mathbb{R}$ and $Q$ be the set of all orthogonal matrices of order $3$ over $\mathbb{R}$. Then,

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A
$P$ is a proper subset of $Q$
B
$Q$ is a proper subset of $P$
C
Neither $P$ is a proper subset of $Q$ nor $Q$ is a proper subset of $P$
D
$P \cap Q = \phi$, the void set

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3 and 4 .Determinants and Matrices

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Types of matrices, Algebra of matrices

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Let $P$ be the set of all non-singular matrices of order $3$ over $\mathbb{R}$ and $Q$ be the set of all orthogonal matrices of order $3$ over $\mathbb{R}$. Then,

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