Inverse of a Matrix

Multiplicative Inverse of a Matrix

For a square matrix A, the inverse is written A-1. When A is multiplied by A-1 the result is the identity matrix I.

Non-square matrices do not have inverses.

AA-1 = A-1A = I

Are C and D inverses?

Requirements to have an Inverse1.The matrix must be square

(same number of rows and columns).2. The determinant of the matrix must not be zero

•Evaluate the following determinant:

Multiply the diagonals, and subtract:   

The computations for 3×3 determinants are messier than for 2×2's. Various methods can be used, but the simplest is probably the following:   

Take a matrix A:

Write down its determinant:

Extend the determinant's grid by rewriting the first two columns of numbers

 

Then multiply along the down-diagonals of 3 numbers:

...and along the up-diagonals of three numbers

 

Add the down-diagonals and subtract the up-diagonals:

Then det(A)= 1.

And simplify

Find the determinant of the following matrix:

 

First convert from the matrix to its determinant, with the extra columns:

 

Then multiply down and up the diagonals:

Then add the down-diagonals, subtract the up-diagonals, and simplify for the final answer: