Question

How to find the inverse of this matrix


`\left[\begin{array}{ccc}1&0\\0&3\\\end{array}\right]`

Answer 1

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) = (1)/(det(M)) *adj(M) = (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]`