If A = [ cc 1 & 2 -1 & -2 ], B = [ cc 2 & a -1 & b ] and if ( A +… - Vidyadip

Question

If $A =\left[\begin{array}{cc}1 & 2 \\ -1 & -2\end{array}\right], B =\left[\begin{array}{cc}2 & a \\ -1 & b\end{array}\right]$ and if $( A + B )^2= A ^2+ B ^2$, find values of $a$ and $b$.

✓

Answer

$
\begin{aligned}
& ( A + B )^2= A ^2+ B ^2 \\
& \therefore( A + B )( A + B )= A ^2+ B ^2 \\
& \therefore A ^2+ AB + BA + B ^2= A ^2+ B ^2 \\
& \therefore AB + BA =0 \\
& \therefore AB =- BA \\
& \therefore\left[\begin{array}{rr}
1 & 2 \\
-1 & -2
\end{array}\right]\left[\begin{array}{rr}
2 & a \\
-1 & b
\end{array}\right]=-\left[\begin{array}{rr}
2 & a \\
-1 & b
\end{array}\right]\left[\begin{array}{rr}
1 & 2 \\
-1 & -2
\end{array}\right] \\
& \therefore\left[\begin{array}{rr}
2-2 & a+2 b \\
-2+2 & -a-2 b
\end{array}\right]=-\left[\begin{array}{rr}
2-a & 4-2 a \\
-1-b & -2-2 b
\end{array}\right] \\
& \therefore\left[\begin{array}{rr}
0 & a+2 b \\
0 & -a-2 b
\end{array}\right]=\left[\begin{array}{rr}
a-2 & 2 a-4 \\
1+b & 2+2 b
\end{array}\right]
\end{aligned}
$
By the equality of matrices, we get
$
\begin{aligned}
& 0=a-2 \ldots \ldots . .(1) \\
& 0=1+b \ldots \ldots . .(2) \\
& a+2 b=2 a-4 . . \\
& -a-2 b=2+2 b
\end{aligned}
$
From equations (1) and (2), we get $a=2$ and $b=-1$
The values of $a$ and $b$ satisfy equations (3) and (4) also. Hence, $a=2$ and $b=-1$.

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