11 0 0
Ex. (7) If A= 5 1 0
that BA = I.
1 3
1
then find matrix B such that AB = I. Verify
Answers
\textbf{Given:}Given:
\begin{gathered}\mathsf{A=\left(\begin{array}{ccc}1&0&0\\5&1&0\\1&3&1\end{array}\right)}\end{gathered}
A=
⎝
⎛
1
5
1
0
1
3
0
0
1
⎠
⎞
\textbf{To find:}To find:
\textsf{The matrix B}The matrix B
\textbf{Solution:}Solution:
\textsf{Consider,}Consider,
\mathsf{AB=I}AB=I
\implies\mathsf{B=A^{-1}}⟹B=A
−1
\begin{gathered}\mathsf{|A|=\left|\begin{array}{ccc}1&0&0\\5&1&0\\1&3&1\end{array}\right|}\end{gathered}
∣A∣=
∣
∣
∣
∣
∣
∣
∣
1
5
1
0
1
3
0
0
1
∣
∣
∣
∣
∣
∣
∣
\mathsf{=1(1-0)-0+0=1\neq\,0}=1(1−0)−0+0=1
=0
\mathsf{A^{-1}\;exists}A
−1
exists
\begin{gathered}\mathsf{adjA=\left(\begin{array}{ccc}(1-0)&-(5-0)&(15-1)\\-(0-0)&(1-0)&-(3-0)\\(0-0)&-(0-0)&(1-0)\end{array}\right)^T}\end{gathered}
adjA=
⎝
⎛
(1−0)
−(0−0)
(0−0)
−(5−0)
(1−0)
−(0−0)
(15−1)
−(3−0)
(1−0)
⎠
⎞
T
\begin{gathered}\mathsf{adjA=\left(\begin{array}{ccc}1&-5&14\\0&1&-3\\0&0&1\end{array}\right)^T}\end{gathered}
adjA=
⎝
⎛
1
0
0
−5
1
0
14
−3
1
⎠
⎞
T
\begin{gathered}\mathsf{adjA=\left(\begin{array}{ccc}1&0&0\\-5&1&0\\14&-3&1\end{array}\right)}\end{gathered}
adjA=
⎝
⎛
1
−5
14
0
1
−3
0
0
1
⎠
⎞
\mathsf{Now,\;A^{-1}=\dfrac{1}{|A|}adjA}Now,A
−1
=
∣A∣
1
adjA
\begin{gathered}\mathsf{A^{-1}=\left(\begin{array}{ccc}1&0&0\\-5&1&0\\14&-3&1\end{array}\right)}\end{gathered}
A
−1
=
⎝
⎛
1
−5
14
0
1
−3
0
0
1
⎠
⎞
\begin{gathered}\implies\boxed{\mathsf{B=\left(\begin{array}{ccc}1&0&0\\-5&1&0\\14&-3&1\end{array}\right)}}\end{gathered}
⟹
B=
⎝
⎛
1
−5
14
0
1
−3
0
0
1
⎠
⎞
\mathsf{BA}BA
\begin{gathered}\mathsf{=\left(\begin{array}{ccc}1&0&0\\-5&1&0\\14&-3&1\end{array}\right)\left(\begin{array}{ccc}1&0&0\\5&1&0\\1&3&1\end{array}\right)}\end{gathered}
=
⎝
⎛
1
−5
14
0
1
−3
0
0
1
⎠
⎞
⎝
⎛
1
5
1
0
1
3
0
0
1
⎠
⎞
\begin{gathered}\mathsf{=\left(\begin{array}{ccc}1+0+0&0+0+0&0+0+0\\-5+5+0&0+1+0&0+0+0\\14-15+1&0-3+3&0+0+1\end{array}\right)}\end{gathered}
=
⎝
⎛
1+0+0
−5+5+0
14−15+1
0+0+0
0+1+0
0−3+3
0+0+0
0+0+0
0+0+1
⎠
⎞
\begin{gathered}\mathsf{=\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right)}\end{gathered}
=
⎝
⎛
1
0
0
0
1
0
0
0
1
⎠
⎞
\mathsf{=I}=I
\mathsf{Hence\;verified}Henceverified