Is there any formula for division of two matrices If yes then what is that n if no - Maths - Matrices

Question

Is there any formula for division of two matrices? If yes then
what is that? n if no then how can we divide two matrix ?

Gursheen kaur. · Accepted Answer

There is no formula for the division of matrices. It can only be solved by Inverse of Matrices method.

The inverse of a matrix A, i.e. A^-1 can be found out by the following way:

Find determinant of A, i.e. |A|.
Find the adjacent matrix of A, i.e. adjA.
Put values of |A|and adjA in the formula:A^-1= Scalar product of reciprocal of |A| and adjA = |A|^-1|adjA = adjA / |A|

Lets take an examples,

1) A is a square matrix ,A =I²,then what is A inverse?

Solution,

A=I²

Thus,

A=I.I

=I

Thus,

Postmultiplying by A^-1 on both sides,

A.A^-1= I. A^-1

Thus, we get,

I=A^-1

Or A^-1=I

$2) L e t A = [\begin{matrix} 1 & 3 & - 2 \\ - 3 & 0 & 1 \\ 2 & 1 & 0 \end{matrix}] A s, w e k n o w t h a t, A = I A [\begin{matrix} 1 & 3 & - 2 \\ - 3 & 0 & 1 \\ 2 & 1 & 0 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] A$

$R_{2} \to R_{2} + 3 R_{1}, R_{3} \to R_{3} - 2 R_{1} [\begin{matrix} 1 & 3 & - 2 \\ 0 & 9 & - 5 \\ 0 & - 5 & 4 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ - 2 & 0 & 1 \end{matrix}] A R_{1} \to R_{1} + \frac{3}{5} R_{3}, R_{2} \to R_{2} + \frac{8}{5} R_{3} [\begin{matrix} 1 & 0 & \frac{- 37}{20} \\ 0 & 1 & \frac{7}{5} \\ 0 & - 5 & 4 \end{matrix}] = [\begin{matrix} 1 & 0 & \frac{\begin{matrix} 3 \end{matrix}}{5} \\ \frac{- 1}{5} & 1 & \frac{8}{5} \\ - 2 & 0 & 1 \end{matrix}] A R_{3} \to R_{3} + 5 R_{2} [\begin{matrix} 1 & 0 & \frac{- 37}{20} \\ 0 & 1 & \frac{7}{5} \\ 0 & 0 & 11 \end{matrix}] = [\begin{matrix} 1 & 0 & \frac{\begin{matrix} 3 \end{matrix}}{5} \\ \frac{- 1}{5} & 1 & \frac{8}{5} \\ - 3 & 5 & 9 \end{matrix}] A [\begin{matrix} 1 & 0 & \frac{- 37}{20} \\ 0 & 1 & \frac{7}{5} \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & \frac{\begin{matrix} 3 \end{matrix}}{5} \\ \frac{- 1}{5} & 1 & \frac{8}{5} \\ \frac{\begin{matrix} - 3 \end{matrix}}{11} & \frac{5}{11} & \frac{9}{11} \end{matrix}] A R_{3} \to \frac{1}{11} R_{3} [\begin{matrix} 1 & 0 & \frac{- 37}{20} \\ 0 & 1 & \frac{7}{5} \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & \frac{\begin{matrix} 3 \end{matrix}}{5} \\ \frac{- 1}{5} & 1 & \frac{8}{5} \\ \frac{\begin{matrix} - 3 \end{matrix}}{11} & \frac{5}{11} & \frac{9}{11} \end{matrix}] A R_{1} \to R_{1} + \frac{37}{20} R_{3}, R_{2} \to R_{2} - \frac{7}{5} R_{3} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} \frac{109}{220} & \frac{37}{44} & \frac{93}{44} \\ \frac{2}{11} & \frac{4}{11} & \frac{5}{11} \\ \frac{\begin{matrix} - 3 \end{matrix}}{11} & \frac{5}{11} & \frac{9}{11} \end{matrix}] A t h e r e f o r e, A^{- 1} = [\begin{matrix} \frac{109}{220} & \frac{37}{44} & \frac{93}{44} \\ \frac{2}{11} & \frac{4}{11} & \frac{5}{11} \\ \frac{\begin{matrix} - 3 \end{matrix}}{11} & \frac{5}{11} & \frac{9}{11} \end{matrix}]$

You can also follow the given link,

The process of finding out the inverse of a matrix is explained with the help of animated video in the following link:

http://cbse.meritnation.com/study-online/study-material/uQtJr3RTOO29PR$ymL4GGQ!!/Hys0$kohdwCWZRaM5ITFVA!!

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Is there any formula for division of two matrices? If yes then what is that? n if no then how can we divide two matrix ?