(B) Given $A = \left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$.Any square matrix $A$ can be uniquely expressed as $A

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(B) Given $A = \left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$.Any square matrix $A$ can be uniquely expressed as $A = P + Q$,where $P = \frac{1}{2}(A + A^T)$ is a symmetric matrix and $Q = \frac{1}{2}(A - A^T)$ is a skew-symmetric matrix.We know that for any symmetric matrix $P$,$P^T = P$,and for any skew-symmetric matrix $Q$,$Q^T = -Q$.Therefore,$P^T - Q^T = P - (-Q) = P + Q = A$.Thus,$P^T - Q^T = A = \left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Mix Examples-Determinants and Matrices

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If $A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$ is expressed as a sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$,then $P^{T}-Q^{T}=$

AP EAMCET 2023 All AP EAMCET

A
$\left[\begin{array}{ccc}8 & -16 & -4 \\ 2 & 8 & 7 \\ 6 & 14 & -16\end{array}\right]$
B
$\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$
C
$\left[\begin{array}{ccc}2 & 4 & -5 \\ 0 & 3 & 7 \\ -3 & 1 & 2\end{array}\right]$
D
$\left[\begin{array}{ccc}1 & 0 & -3/2 \\ 2 & 3/2 & 1/2 \\ -5/2 & 7/2 & 1\end{array}\right]$

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Mix Examples-Determinants and Matrices

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If the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{bmatrix}$ satisfies the equation $A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ for some real numbers $\alpha$ and $\beta$,then $\beta - \alpha$ is equal to ........ .

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If $A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$ is expressed as a sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$,then $P^{T}-Q^{T}=$

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Let $P=\begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{bmatrix}$ and $I$ be the identity matrix of order $3$. If $Q=[q_{ij}]$ is a matrix such that $P^{50}-Q=I$,then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

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