Question
Find $a, b, c$ and $d$ if $3\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{cc}4 & a+b \\ c+d & 3\end{array}\right]+\left[\begin{array}{cc}a & 6 \\ -1 & 2 d\end{array}\right]$
✓
Answer
Given
$
\begin{aligned}
& 3\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]=\left[\begin{array}{cc}
4 & a+b \\
c+d & 3
\end{array}\right]+\left[\begin{array}{cc}
a & 6 \\
-1 & 2 d
\end{array}\right] \\
& \Rightarrow\left[\begin{array}{ll}
3 a & 3 b \\
3 c & 3 d
\end{array}\right]=\left[\begin{array}{cc}
4+a & a+b+6 \\
c+d-1 & 3+2 d
\end{array}\right]
\end{aligned}
$
Comparing the corresponding elements:
$
\begin{aligned}
& 3 a=4+a \\
& \Rightarrow 3 a-a=4 \\
& \Rightarrow 2 a=4 \\
& \therefore a=2 \\
& 3 b=a+b+6 \\
& \Rightarrow 3 b-b=2+6 \\
& \Rightarrow 2 b=8 \\
& \therefore b=4 \\
& 3 d=3+2 d \\
& \Rightarrow 3 d-2 d=3 \\
& \therefore d=3 \\
& 3 c=c+d-1 \\
& \Rightarrow 3 c-c=3-1 \\
& 2 c=2 \\
& \Rightarrow c=1
\end{aligned}
$
Hence $a=2, b=4, c=1, d=3$.
$
\begin{aligned}
& 3\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]=\left[\begin{array}{cc}
4 & a+b \\
c+d & 3
\end{array}\right]+\left[\begin{array}{cc}
a & 6 \\
-1 & 2 d
\end{array}\right] \\
& \Rightarrow\left[\begin{array}{ll}
3 a & 3 b \\
3 c & 3 d
\end{array}\right]=\left[\begin{array}{cc}
4+a & a+b+6 \\
c+d-1 & 3+2 d
\end{array}\right]
\end{aligned}
$
Comparing the corresponding elements:
$
\begin{aligned}
& 3 a=4+a \\
& \Rightarrow 3 a-a=4 \\
& \Rightarrow 2 a=4 \\
& \therefore a=2 \\
& 3 b=a+b+6 \\
& \Rightarrow 3 b-b=2+6 \\
& \Rightarrow 2 b=8 \\
& \therefore b=4 \\
& 3 d=3+2 d \\
& \Rightarrow 3 d-2 d=3 \\
& \therefore d=3 \\
& 3 c=c+d-1 \\
& \Rightarrow 3 c-c=3-1 \\
& 2 c=2 \\
& \Rightarrow c=1
\end{aligned}
$
Hence $a=2, b=4, c=1, d=3$.
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