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Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMatrices1 Mark
MCQ
If $A =\left[\begin{array}{ll}1 & a \\ 0 & 1\end{array}\right]$ then $A ^4$ equals to :
  • A
    $\left[\begin{array}{cc}1 & a^4 \\ 0 & 1\end{array}\right]$
  • B
    $\left[\begin{array}{cc}4 & 4 a \\ 0 & 4\end{array}\right]$
  • C
    $\left[\begin{array}{cc}4 & a^4 \\ 0 & 4\end{array}\right]$
  • ✓
    $\left[\begin{array}{cc}1 & 4 a \\ 0 & 1\end{array}\right]$
✓

Answer

Correct option: D.
$\left[\begin{array}{cc}1 & 4 a \\ 0 & 1\end{array}\right]$
(D)
$
\begin{aligned}
A^2 & =A \cdot A=\left[\begin{array}{ll}
1 & a \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
1 & a \\
0 & 1
\end{array}\right] \\
A^2 & =\left[\begin{array}{cc}
1 & 2 a \\
0 & 1
\end{array}\right] \\
A^4=A^2 A^2 & =\left[\begin{array}{cc}
1 & 2 a \\
0 & 1
\end{array}\right]\left[\begin{array}{cc}
1 & 2 a \\
0 & 1
\end{array}\right]=\left[\begin{array}{cc}
1+0 & 2 a+2 a \\
0+0 & 0+1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 & 4 a \\
0 & 1
\end{array}\right]
\end{aligned}
$
Hence correct option is (D).

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