How to find the inverse of a matrix (and the determinant)

This video shows you how to calculate the inverse of a 2x2 square matrix. It also shows you how to calculate the determinant, since this is part of the process required to find the inverse.
Transcript:
In today’s video I’m going to show you how to find the inverse of a square matrix.
In this question we are given a two-by-two square matrix labelled P, and we are asked to find the inverse, which is P raised to the power minus one.
We calculate the inverse using a two-step process,
so let’s begin by moving the original matrix to the top of the screen.
And now we are ready to start with Step 1.
So the first thing we need to do is find the determinant of the original matrix.
If we consider a general matrix with the elements a, b, c and d, then the determinant is a times d minus b times c.
So what we are doing is multiplying the two diagonals and then subtracting the products in the order specified.
So in our matrix, the leading diagonal gives three times four minus one times two, which equals twelve minus two, which is 10.
So we have completed Step 1 of the calculation, and can conclude that the determinant of matrix P is ten.
So we now move on to Step 2 of the calculation, which will use the determinant we’ve just calculated.
This is because the inverse matrix is one over the determinant multiplied by the following rearranged matrix.
If we look back at the general matrix with the elements a, b, c and d, you will notice that the two elements on the leading diagonal have swapped places, and the elements on the other diagonal have changed their sign.
The inverse of matrix P is therefore one divided by the determinant multiplied by the following rearranged version of P, where the elements on the leading diagonal have swapped places, and the two elements on the other diagonal have swapped sign.
So we can conclude that the inverse of matrix P is as follows.
We could present this in an alternative way by multiplying each element inside the matrix by one tenth,
but this would create a matrix that contains fractions and it is usually best to leave whole numbers inside the matrix and keep any fractions outside.
So it’s not necessary to try to simply this matrix any further and we have therefore found the inverse of P as required.

[@DavidWrightUK]

If you would like to support my channel you can now buy me a coffee here: www.buymeacoffee.com/DavidWrightUK ❤ 🙏🏼 😎