Question

what is range of matrix A ??​

Answer 1

$\begin{gathered}\tt A=\left[\begin{array}{ccc}1&sinx&1\\ - sinx&1&sinx\\1&-sinx&1\end{array}\right]\end{gathered}$

$:\implies \tt \: |A| = \begin{array}{|ccc|}</p><p>1 & sinx & 1 \\</p><p>-sinx & 1 & sinx \\</p><p>1 &-sinx & 1\\</p><p>\end{array}$

$:\implies \tt \: |A| = 1(1 + {sin}^{2} x) - sinx( - sinx - sinx) + 1( {sin}^{2} x - 1)$

$:\implies \tt \: |A| = 1 + {sin}^{2} x + 2 {sin}^{2} x + {sin}^{2} x - 1$

$:\implies \tt \: |A| = 4 {sin}^{2} x$

Hence, Two cases arises,

Case 1

$:\implies \tt \: when \: x \: \ne \: n\pi$

$:\implies \tt \: |A| \: \ne \: 0$

$:\implies \boxed{ \pink{\tt \: Range \: of \: matrix\: A \: is \: 3}}$

Case 2

$:\implies \tt \: when \: x \: = \: n\pi$

$:\implies \tt \: |A| = {sin}^{2} n\pi \: = 0$

$:\implies \tt \: Range \: of \: A \: \ne \: 3$

Consider submatrix of order 2 × 2,

$\begin{vmatrix} 1 & sinx \\ - sinx & 1 \end{vmatrix}$

$:\implies \tt \: 1 + {sin}^{2} x$

$:\implies \tt \: 1 + {sin}^{2} n\pi$

$:\implies \tt \: 1 + 0$

$:\implies \tt \: 1 \: \ne \: 0$

$:\implies \boxed{ \pink{\tt \: Range \: of \: matrix\: A \: is \: 2}}$

$\begin{gathered}\begin{gathered}\bf Range \: of \: matrix\: A \: is -  \begin{cases} &\sf{3 \: if \: x \: \ne \: n\pi} \\ &\sf{2 \: if \: x \: = \: n\pi} \end{cases}\end{gathered}\end{gathered}$