(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 $|A|^2 = 1$,whic

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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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$\begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} \begin{bmatrix} 2 & 1 & -1 \end{bmatrix} = $

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Given $A = \begin{bmatrix} \sqrt{3} & 1 & -1 \\ 2 & 3 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & \sqrt{5} & 1 \\ -2 & 3 & \frac{1}{2} \end{bmatrix}$,find $A + B = \dots \dots \dots$

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$A$ square matrix $[a_{ij}]_{n \times n}$ is an upper triangular matrix if:

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If two matrices $A$ and $B$ are of order $p \times q$ and $r \times s$ respectively,they can be subtracted only if:

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Let $A = [a_{ij}]$ be a real matrix of order $3 \times 3$,such that $a_{i1} + a_{i2} + a_{i3} = 1$,for $i = 1, 2, 3$. Then,the sum of all the entries of the matrix $A^3$ is equal to:

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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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