Question 8.4.3: Using the Quick Method to Find a Multiplicative Inverse Find...
Using the Quick Method to Find a Multiplicative Inverse
Find the multiplicative inverse of
A=\left[\begin{array}{rr}-1 & -2 \\3 & 4\end{array}\right].
This is the given matrix. We’ve designated the elements a, b, c, and d.
A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\-c & a\end{array}\right] This is the formula for the inverse of \left[\begin{array}{ll}a & b \\ c & d\end{array}\right].
=\frac{1}{(-1)(4)-(-2)(3)}\left[\begin{array}{rr}4 & -(-2) \\-3 & -1\end{array}\right] Apply the formula with a = –1, b = –2, c = 3, and d = 4.
=\frac{1}{2}\left[\begin{array}{rr}4 & 2 \\-3 & -1\end{array}\right] Simplify.
=\left[\begin{array}{rr}2 & 1 \\-\frac{3}{2} & -\frac{1}{2}\end{array}\right] Perform the scalar multiplication by multiplying each element in the matrix by \frac{1}{2}.
The inverse of A=\left[\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right] is A^{-1}=\left[\begin{array}{rr}2 & 1 \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right].
We can verify this result by showing that A A^{-1}=I_2 and A^{-1} A=I_2.