Question:

Show that 
(i)[5167]\begin{bmatrix}5&-1\\6&7\end{bmatrix}[2134]\begin{bmatrix}2&1\\3&4\end{bmatrix}[2134][5167]\neq \begin{bmatrix}2&1\\3&4\end{bmatrix}\begin{bmatrix}5&-1\\6&7\end{bmatrix}

(ii)[123010110]\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}[110011234]\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}\neq  [110011234]\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}[123010110]\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}

Updated On: Aug 30, 2023
Hide Solution
collegedunia
Verified By Collegedunia

Solution and Explanation

(i)[5167]\begin{bmatrix}5&-1\\6&7\end{bmatrix}[2134]\begin{bmatrix}2&1\\3&4\end{bmatrix}

=[5(2)1(3)5(1)1(4)6(2)+7(3)6(1)+7(4)]\begin{bmatrix}5(2)-1(3)&5(1)-1(4)\\6(2)+7(3)&6(1)+7(4)\end{bmatrix}

=[1035412+216+28]\begin{bmatrix}10-3&5-4\\12+21&6+28\end{bmatrix}=[713334]\begin{bmatrix}7&1\\33&34\end{bmatrix}

[2134]\begin{bmatrix}2&1\\3&4\end{bmatrix}[2134]\begin{bmatrix}2&1\\3&4\end{bmatrix}

=[2(5)+1(6)2(1)+1(7)3(5)+4(6)3(1)+4(7)]\begin{bmatrix}2(5)+1(6)&2(-1)+1(7)\\3(5)+4(6)&3(-1)+4(7)\end{bmatrix}

=[10+62+715+243+28]\begin{bmatrix}10+6&-2+7\\15+24&-3+28\end{bmatrix}

=[[1653925]\begin{bmatrix}16&5\\39&25\end{bmatrix}

\therefore [5167]\begin{bmatrix}5&-1\\6&7\end{bmatrix}[2134]\begin{bmatrix}2&1\\3&4\end{bmatrix}[2134]\begin{bmatrix}2&1\\3&4\end{bmatrix}[5167]\begin{bmatrix}5&-1\\6&7\end{bmatrix}


(ii)[123010110]\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}[110011234]\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}

=[1(1)+2(0)+3(2)1(1)+2(1)+3(3)1(0)+2(1)+3(4)0(1)+1(0)+0(2)0(1)+1(1)+0(3)0(0)+1(1)+0(4)1(1)+1(0)+0(2)1(1)+1(1)+0(3)1(0)+1(1)+0(4)]\begin{bmatrix}1(-1)+2(0)+3(2)&1(1)+2(-1)+3(3)&1(0)+2(-1)+3(4)\\0(-1)+1(0)+0(2)&0(1)+1(-1)+0(3)&0(0)+1(1)+0(4)\\1(-1)+1(0)+0(2)&1(1)+1(-1)+0(3)&1(0)+1(1)+0(4)\end{bmatrix}

=[5814011101]\begin{bmatrix}5&8&14\\0&-1&1\\ -1&0&1\end{bmatrix}

[110011234]\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}[123010110]\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}

=[1(1)1(0)+0(1)1(2)+1(1)+0(1)1(3)+1(0)+0(0)0(1)+(1(0)+1(1)0(2)+(1)(1)+1(1)0(3)+(1)(0)+1(0)2(1)+3(0)+4(1)2(2)+3(1)+4(1)2(3)+3(0)+4(0)]\begin{bmatrix}-1(1)1(0)+0(1)&-1(2)+1(1)+0(1)&-1(3)+1(0)+0(0)\\0(1)+(-1(0)+1(1)&0(2)+(-1)(1)+1(1)&0(3)+(-1)(0)+1(0)\\2(1)+3(0)+4(1)&2(2)+3(1)+4(1)&2(3)+3(0)+4(0)\end{bmatrix}

=[1131006116]\begin{bmatrix}-1&-1&-3\\1&0&0\\ 6&11&6\end{bmatrix}

\therefore  [123010110]\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}[110011234]\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}≠ [110011234]\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}[123010110]\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}

Was this answer helpful?
0
0

Top Questions on Matrices

View More Questions