Question

Which among the below matrices has the inverse A^-1=\(\begin{bmatrix}1&-\frac{5}{8}\\0&\frac{1}{8}\end{bmatrix}\)

(a) \(\begin{bmatrix}1&5\\0&8\end{bmatrix}\)

(b) \(\begin{bmatrix}1&5\\-1&8\end{bmatrix}\)

(c) \(\begin{bmatrix}1&5\\0&16\end{bmatrix}\)

(d) \(\begin{bmatrix}1&8\\0&8\end{bmatrix}\)

I had been asked this question by my school principal while I was bunking the class.

My question comes from Invertible Matrices in division Matrices of Mathematics – Class 12

Answer 1

Correct answer is (a) \(\begin{bmatrix}1&5\\0&8\end{bmatrix}\)

Best explanation: Consider the matrix A=\(\begin{bmatrix}1&5\\0&8\end{bmatrix}\)

Using the elementary column operations, we write A=AI

\(\begin{bmatrix}1&5\\0&8\end{bmatrix}\)=A\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)

Applying C2→C2-5C1

\(\begin{bmatrix}1&0\\0&8\end{bmatrix}\)=A\(\begin{bmatrix}1&-5\\0&1\end{bmatrix}\)

Applying C2→C2/8

\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)=A\(\begin{bmatrix}1&-\frac{5}{8}\\0&\frac{1}{8}\end{bmatrix}\)

A^-1=\(\begin{bmatrix}1&-\frac{5}{8}\\0&\frac{1}{8}\end{bmatrix}\).