Question

Maharashtra State Board
Class 11
Mathematics and statistics part 1
Chapter - 4 : Determinants and Matrices
Please help me with this question!

Answer 1

HENCE PROVED

I hope it will be helpful for you ✌️✌️

Thank you ☺️☺️

Answer 2

Answer:

The given matrices are

$A=\begin{bmatrix}1&2\\ 3&5\end{bmatrix}$

$B=\begin{bmatrix}0&4\\ 2&-1\end{bmatrix}$

We need to prove AB≠BA

$AB=\begin{bmatrix}1\cdot \:0+2\cdot \:2&1\cdot \:4+2\left(-1\right)\\ 3\cdot \:0+5\cdot \:2&3\cdot \:4+5\left(-1\right)\end{bmatrix}=\begin{bmatrix}4&2\\ 10&7\end{bmatrix}$

$BA=\begin{bmatrix}0\cdot \:1+4\cdot \:3&0\cdot \:2+4\cdot \:5\\ 2\cdot \:1+\left(-1\right)\cdot \:3&2\cdot \:2+\left(-1\right)\cdot \:5\end{bmatrix}=\begin{bmatrix}12&20\\ -1&-1\end{bmatrix}$

$\begin{bmatrix}4&2\\ 10&7\end{bmatrix}\neq \begin{bmatrix}12&20\\ -1&-1\end{bmatrix}$

Hence proved AB≠BA.

We need to prove that |AB|=|A||B|.

We know that

$\begin{bmatrix}a\:&\:b\:\\ c\:&\:d\:\end{bmatrix}\:=\:ad-bc$

$|A|=1\cdot \:5-2\cdot \:3=-1$

$|B|=0\cdot \left(-1\right)-4\cdot \:2=-8$

$|AB|=4\cdot \:7-2\cdot \:10=8$

$|A||B|=(-1)(-8)=8=|AB|$

Hence proved |AB|=|A||B|.