Question

If A is nonsingular, then det A−1= 1/(det A). Verify this
theorem for 2 3 2 matrices.

Answer 1

We have to prove

`det(A^(-1))=(1)/(det(A))`

In general

`A.A^(-1)=I\\\\|A.A^(-1)|=|\,I\,|\\\\|\,I\,|=1\\\\\implies|A|.|(A^(-1))|=1\\\\|A^(-1)|=(1)/(|A|)`

Consider a 2 x 2 matrix

`A=\left[\begin{array}{cc}1&2\\4&5\end{array}\right] \\\\|A|=(5)(1)-(2)(4)\\\\|A|=-3---(1)`

Now we find A⁻¹

`A=\left[\begin{array}{cc}1&2\\4&5\end{array}\right] \\\\|A|=-3\\\\Adj(A)=\left[\begin{array}{cc}5&-2\\-4&1\end{array}\right] \\\\\\A^(-1)=(Adj(A))/(|A|)\\\\A^(-1)=\left[\begin{array}{cc}(5)/(-3)&(-2)/(-3)\\(-4)/(-3)&(1)/(-3)\end{array}\right] \\\\\\|A^(-1)|=((5)/(-3))((1)/(-3))-((-4)/(-3))((-2)/(-3))\\\\|A^(-1)|=(5)/(9)-(8)/(9)\\\\|A^(-1)|=(-1)/(3)`

Hence proved.