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Prove that inverse of Matrix $A=${a b c d} (2x2 matrix!) $A^{-1}$=$1/(ad-bc)${d -b -c a} (its also 2x2 matrix!) by solving a set of 4 linear equations in 4 unknowns.
This is what my homework asked. I know how to find inverse of $A=${a b c d} matrix by adding Identity matrix next to the matrix and making original matrix identity. But i didn't understand by solving a set of 4 linear equations in 4 unknowns.part
By this sentence, do I have to use other method than i know to find inverse of it ? If so what kind of method teacher might expect ?
Second question: how to write a matrix in this site?

Andy
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  • To see how to write matrices, check the only answer (so far) below. Click the edit button to see the code. Instead of pmatrix you can also use bmatrix. – Git Gud Oct 05 '14 at 19:02

2 Answers2

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You are asked to solve the system of linear equations $$\left(\begin{array}{cc}a &b\\ c& d\end{array}\right) \left(\begin{array}{cc}x& z\\ y& t\end{array}\right)=\left(\begin{array}{cc}1&0\\ 0& 1 \end{array}\right)$$ on the unknowns $x,y,z,t.$ That is:

$$\left\{\begin{array}{r}ax+by=1 \\ cx+dy=0\\az+dt=0\\cz+dt=1\end{array}\right.$$

Note that it is equivalent to solve the following two linear systems:

$$\left\{\begin{array}{r}ax+by=1 \\ cx+dy=0\end{array}\right.$$ and

$$\left\{\begin{array}{r}az+dt=0\\cz+dt=1\end{array}\right.$$

mfl
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You are seeking $A^{-1}$. Put $$ A^{-1} := \begin{pmatrix} a_1 & a_2 \\ a_3 & a_4 \end{pmatrix}. $$ You want $AA^{-1} = I$ and $A^{-1}A = I$. Now, by If $AB = I$ then $BA = I$, it is sufficient to solve, say, $AA^{-1} = I$, i.e. $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a_1 & a_2 \\ a_3 & a_4 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, $$ or equivalently, you must solve the system of $4$ linear equations in the $4$ unkowns $a_1, a_2, a_3, a_4$ given by \begin{align} a a_1 + ba_3 = 1, \\ a a_2 + ba_4 = 0, \\ c a_1 + da_3 = 1, \\ c a_2 + da_4 = 0. \end{align} As pointed out in another answer here, this amounts to solving two independent systems of $2$ equations in $2$ unknowns.

To write matrices on this site you can click "edit" on this answer and you will see the Latex code.

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