Question

How to find the inverse of this matrix

Answer 1

Answer:

Here we have the matrix:

$M = \left[\begin{array}{ccc}1&0\\0&3\end{array}\right]$

And we want to find its inverse.

The inverse of a 2x2 matrix A is:

(1/det(A))*adj(A)

where det(A) is the determinant of the matrix.

Such that for a matrix:

$A = \left[\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right]$

The determinant is:

det(A) = a₁₁*a₂₂ - a₁₂*a₂₁

in the case of our matrix M, the determinant is:

det(M) = 1*3 - 0*0 = 3

and adj(A) is a transposition along the diagonal, and for the other elements, we just change its sign.

Then for our matrix A we would have:

$adj(A) = \left[\begin{array}{ccc}a_{22}&-a_{12}\\-a_{21}&a_{11}\end{array}\right]$

Then for our matrix M, we have:

$adj(M) = \left[\begin{array}{ccc}3&-0\\-0&1\end{array}\right]$

Then the inverse of the matrix M is:

$M^{-1} = \frac{1}{det(M)} *adj(M) = \frac{1}{3}\left[\begin{array}{ccc}3&0\\0&1\end{array}\right] = \left[\begin{array}{ccc}1&0\\0&1/3\end{array}\right]$