You need to calculate the determinant of the matrix as an initial step. If the determinant is 0, then your work is finished, because the matrix has no inverse. The determinant of matrix M can be represented symbolically as det M. Transposing means reflecting the matrix about the main diagonal, or equivalently, swapping the i,j th element and the j,i th. When you transpose the terms of the matrix, you should see that the main diagonal from upper left to lower right is unchanged. Notice the colored elements in the diagram above and see where the numbers have changed position.

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Do you know what the inverse of a 3x3 matrix is and how to find it? This lesson goes over these and related concepts necessary for finding the inverse of a sample 3x3 matrix. Setting up the Problem Why would you ever need to find the inverse of a 3x3 matrix? Well, matrices and inverse matrices have lots of applications in geometry, the sciences, and especially computer science. To find the inverse of a 3x3 matrix, we first have to know what an inverse is. Mathematically, this definition is pretty simple.

M is just our original matrix. M raised to the power of -1 is the mathematical symbol for the inverse matrix of M. And finally, I is the identity matrix, which has 1s on the main diagonal and 0s everywhere else.

Simple enough in concept, right? But how do you calculate a matrix inverse? Determinant The determinant of a matrix is a single number that is characteristic of that matrix. You can find the determinant using several methods.

Because this method reduces the number of calculations if you have any zeros in your matrix. Since we have both a zero in our matrix and would like to reduce the number of calculations, this method will work well for us.

Cofactors are the determinants of the submatrix of a matrix element that does not include the rows or column of that element. For example, the cofactor of the matrix element of M in the first row and first column will be the determinant of the submatrix that does not include any elements from either the first row 1, 2, 3 or first column 1, 0, 1.

The first row, first column element expansion times its cofactor looks like this: Continuing our expansion along the first column, we will have the first column 1, 0, 1 , second row 0, 1, -2 element times the cofactor.

But wait! That element is equal to 0, and anything multiplied by 0 is just 0. This saves us a step, which is what we wanted in the first place. This means the determinant of our matrix is equal to 2. A lot of work for a small number, but we are making progress.

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## Inverse of a 3 by 3 Matrix

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## Conjugate transpose

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