Maharashtra Board Solutions for Chapter: Determinants and Matrices, Exercise 8: EXERCISE 4.7

Find the second row first column element of $A^{\mathrm{T}}$, if $A=\left[\begin{array}{cc}1& 3\\ -4& 5\end{array}\right]$

Find third-row second column element of $A^T$ if $A=\left[\begin{array}{ccc}2& -6& 1\\ -4& 0& 5\end{array}\right]$

If $A=[ a_{ij}{]}_{3\times 3}$ where $a_{ij}= 2(i-j)$ Find$A \& A^T$. State whether $A \& A^T$are symmetric or skew symmetric matrices ?

If$$$A=\left[\begin{array}{ccc}7& 3& 0\\ 0& 4& -2\end{array}\right], B=\left[\begin{array}{ccc}0& -2& 3\\ 2& 1& -4\end{array}\right]$ then find $5A^T-5B^T$.

Express the following matrix as the sum of a symmetric and a skew symmetric matrix:$\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right]$,

If $A=\left[\begin{array}{cc}2& -1\\ 3& -2\\ 4& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 3& -4\\ 2& -1& 1\end{array}\right]$, then verify that $(AB{)}^T=B^T{A}^T$.

If $A=\left[\begin{array}{ll}2& -1\\ 3& -2\\ 4& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 3& -4\\ 2& -1& 1\end{array}\right]$, then verify that ${\left(BA\right)}^T=A^T{B}^T$.

If $A=\left[\begin{array}{cc}\mathrm{cos\alpha }& \mathrm{sin\alpha }\\ -\mathrm{sin\alpha }& \mathrm{cos\alpha }\end{array}\right]$, then show that $A^{T}A=I$, where $I$ is the unit matrix of order $2$.