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Matrices — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsMatrices4 Marks
Question
Find the matrix $B$ if $A=\left[\begin{array}{ll}4 & 1 \\ 2 & 3\end{array}\right]$ and $A^2=A+2 B$
✓

Answer

$
\begin{aligned}
& A =\left[\begin{array}{ll}
4 & 1 \\
2 & 3
\end{array}\right] \\
& \text { Let } B =\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \\
& A ^2= A \times A =\left[\begin{array}{ll}
4 & 1 \\
2 & 3
\end{array}\right]\left[\begin{array}{ll}
4 & 1 \\
2 & 3
\end{array}\right] \\
& =\left[\begin{array}{cc}
16+2 & 4+3 \\
8+6 & 2+9
\end{array}\right] \\
& =\left[\begin{array}{cc}
18 & 7 \\
14 & 11
\end{array}\right] \\
& A +2 B =\left[\begin{array}{ll}
4 & 1 \\
2 & 3
\end{array}\right]+2\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \\
& =\left[\begin{array}{cc}
4 & 1 \\
2 & 3
\end{array}\right]+\left[\begin{array}{cc}
2 a \\
2 c & 3+2 d
\end{array}\right] \\
& =\left[\begin{array}{cc}
4+2 a & 1+2 b \\
2+2 c & 3+2 d
\end{array}\right] \\
& \because A 2= A +2 B \\
& \therefore\left[\begin{array}{cc}
18 & 7 \\
14 & 11
\end{array}\right]=\left[\begin{array}{cc}
4+2 a & 1+2 b \\
2+2 c & 3+2 d
\end{array}\right]
\end{aligned}
$
Comparing the corresponding elements
$
\begin{aligned}
& 4+2 a=18 \\
& \Rightarrow 2 a=18-4=14 \\
& \therefore a=7 \\
& 1+2 b=7 \\
& \Rightarrow 2 b=7-1=6 \\
& \therefore b=3 \\
& 2+2 c=14 \\
& \Rightarrow 2 c=14-2=12 \\
& \therefore c=6 \\
& 3+2 d=11 \\
& \Rightarrow 2 d=11-3=8 \\
& \therefore d=4
\end{aligned}
$
Hence $a=7, b=3, c=6, d=4$
$
\therefore B=\left[\begin{array}{ll}
7 & 3 \\
6 & 4
\end{array}\right] \text {. }
$

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