Question

matrix A = [ -1, 3,5,1,-3, -5, -1, 3,5] show that A2 =A​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\begin{gathered}\rm :\longmapsto\:\sf A=\left[\begin{array}{ccc} - 1&3&5\\1& - 3&-5\\ - 1&3&5\end{array}\right]\end{gathered}$

Consider,

$\rm :\longmapsto\: {A}^{2}$

$\rm \: = \: \: A \times A$

$\: \rm \: = \:\begin{gathered}\left[\begin{array}{ccc} - 1&3&5\\1& - 3&-5\\ - 1&3&5\end{array}\right]\end{gathered}\:\begin{gathered}\left[\begin{array}{ccc} - 1&3&5\\1& - 3&-5\\ - 1&3&5\end{array}\right]\end{gathered}$

$\: \rm \: = \:\:\begin{gathered}\left[\begin{array}{ccc} - 1 + 3 - 5& - 3 - 9 + 15& - 5 - 15 + 25\\ - 1 - 3 + 5& 3 + 9 - 15&5 + 15 - 25\\1 + 3 - 5& - 3 - 9 + 15& - 5 - 15 + 25\end{array}\right]\end{gathered}$

$\: \rm \: = \:\:\begin{gathered}\left[\begin{array}{ccc} - 1&3&5\\1& - 3&-5\\ - 1&3&5\end{array}\right]\end{gathered}$

$\: \rm \: = \:A$

Hence,

$\bf\implies \: {A}^{2} = A$

Additional Information :-

1. If A and B are two matrices, then AB is possible only when number of columns if matrix A is equals to number of rows of matrix B.

2. If A and B are two matrices, then matrix addition or subtraction is possible only when A and B are of same order.

3. Matrix multiplication may or may not be Commutative.

4. Matrix multiplication is Associative, i.e. A(BC) = (AB)C

5. Matrix multiplication is Distributive, i.e. A(B + C) = AB + AC