Question

Find the inverse of the matrix A=\(\begin{bmatrix}8&5\\4&1\end{bmatrix}\).

(a) \(\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\)

(b) \(\begin{bmatrix}\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\)

(c) \(\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&\frac{2}{3}\end{bmatrix}\)

(d) \(\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\-\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\)

I have been asked this question in an online quiz.

The origin of the question is Determinants topic in portion Determinants of Mathematics – Class 12

Answer 1

Correct option is (a) \(\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\)

Easiest explanation: Give that, A=\(\begin{bmatrix}8&5\\4&1\end{bmatrix}\)

adj A=\(\begin{bmatrix}1&-5\\-4&8\end{bmatrix}\)

|A|=8×1-(-5)×(-4)=8-20=-12

A^-1=\(\frac{1}{|A|}\) adj A=\(\frac{1}{-12} \begin{bmatrix}1&-5\\-4&8\end{bmatrix}\)=\(\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\).