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Matrices — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsMatrices4 Marks
Question
Find the values of $a, b, c$ and $d$ if $\left[\begin{array}{ll}a+b & 3 \\ 5+c & a b\end{array}\right]=\left[\begin{array}{cc}6 & d \\ -1 & 8\end{array}\right]$
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Answer

$
\left[\begin{array}{ll}
a+b & 3 \\
5+c & a b
\end{array}\right]=\left[\begin{array}{cc}
6 & d \\
-1 & 8
\end{array}\right]
$
Comparing the corresponding terms, we get.
$
\begin{aligned}
& 3=d \Rightarrow d=3 \\
& \Rightarrow 5+c=-1 \\
& \Rightarrow c=-1-5 \\
& \Rightarrow c=-6 \\
& a+b=6 \text { and } a b=8 \\
& \therefore(a-b)^2=(a+b)^2-4 a b \\
& =(6)^2-4 \times 8 \\
& =36-32 \\
& =4 \\
& =( \pm 2)^2 \\
& \therefore a-b, b= \pm 2 \\
& \text { (i) If } a-b=2 \\
& a+b=6
\end{aligned}
$
Adding, we get
$
\begin{aligned}
& 2 a=8 \\
& \Rightarrow a=4 \\
& a+b=6 \\
& \Rightarrow 4+b=6 \\
& \Rightarrow b=6-4=2 \\
& \therefore a=4, b=2 \\
& \text { (ii) If } a-b=-2 \\
& a+b=6
\end{aligned}
$
Adding, we get,
$
\begin{aligned}
& 2 a=4 \\
& \Rightarrow a=\frac{4}{2}=2 \\
& a+b=6 \\
& \Rightarrow 2+b=6 \\
& \Rightarrow b=6-2=4 \\
& \therefore a=2, b=4
\end{aligned}
$
Hence $a=4, b=2$, or $a=2, b=4$
$
c=-6 \text { and } d=3
$

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