Let A = [3 5; 1 2]. What is the entry in row 1, column 2 of A⁻¹?

Learn / Library / Linear algebra

Undoing a small matrix

Linear algebraEasy

Let $A = [3152]$ . What is the entry in row $1$ , column $2$ of $A^{- 1}$ ?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

For a $2 \times 2$ matrix, swap the diagonal entries, change the sign of the other $2$ entries, and divide by the determinant.

Solution

The determinant is $det (A) = 3 \times 2 - 5 \times 1 = 1$ .
The formula gives $A^{- 1} = [2 - 1 - 5 3]$ divided by $1$ , and indeed $A$ times this matrix is the identity.
So the entry in row $1$ , column $2$ of $A^{- 1}$ is $- 5$ .

Answer: $- 5$