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DETERMINANT OF A MATRIX
The matrix determinant on the other hand is one number associated that can be obtained if a matrix is known.The determinant is most often used to find the nature of solution of the system of linear equations defined by the matrix. The determinant of a 2 by 2 matrix is:

Z = ⎡⎢⎣ a b     c d ⎤⎥⎦

Thus, the det(Z) will always be equal to a × d − c × b. There is nothing too complex about taking the determinant of a two by two matrix. Things get a little more complicated when trying to find the determinant of a 3 x 3 matrix; However, if one is familiar with cross products, this should be fairly simple. Taking the determinant of a three by three matrix is exactly the same as taking the cross product of 2 three dimensional vectors, only this time, replacing the ijk row with the first row of the original matrix. Assume a generic matrix A

A=⎡⎢⎢⎢⎣ a11 a12 a13       a21 a22 a23       a31 a32 a33 ⎤⎥⎥⎥⎦

det(A) = a11(a22a33 − a32a23) − a12(a21a33 − a31a23) + a13(a21a32 − a31a22)

Once again, the determinant of the matrix is a scalar number, not another matrix.


INVERSE OF A MATRIX
The inverse or reciprocal of a square matrix is denoted by A − 1. One interesting fact is a square matrix A times its own inverse A − 1, results in the identity matrix. A square matrix has an inverse if and only if its determinant is not equal to zero. If the determinant is non-zero, then the matrix is an invertible or non singular matrix. If the determinant is zero, then the matrix is a singular or noninvertible matrix.
The inverse of a 2 × 2 matrix is computed as:

A = ⎡⎢⎣ a b     c d ⎤⎥⎦

The next step is to find the determinant of matrix A. Once this has been done, the inverse can be constructed by switching a and d, multiplying c and b by negative one, and finally dividing the matrix by the reciprocal of the det(A):

A − 1 = (1)/(a × d − c × b)⎡⎢⎣ d  − b      − c a ⎤⎥⎦

Taking the inverse of a 3 × 3 involves a little more work but it is indeed possible to do by hand. A general example is shown below. Compute the determinant of the 3 × 3 matrix A, and multiply it by a series of 2 × 2 subdeterminants. It is easiest to understand by observing the pattern from an example:

A=⎡⎢⎢⎢⎣ a11 a12 a13       a21 a22 a23       a31 a32 a33 ⎤⎥⎥⎥⎦

A − 1 = (1)/(det(A))⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣     ⎡⎢⎣ a22 a23     a32 a33 ⎤⎥⎦ ⎡⎢⎣ a13 a12     a33 a32 ⎤⎥⎦ ⎡⎢⎣ a12 a13     a22 a23 ⎤⎥⎦       ⎡⎢⎣ a23 a21     a33 a31 ⎤⎥⎦ ⎡⎢⎣ a11 a13     a31 a33 ⎤⎥⎦ ⎡⎢⎣ a13 a11     a23 a21 ⎤⎥⎦       ⎡⎢⎣ a21 a22     a31 a32 ⎤⎥⎦ ⎡⎢⎣ a12 a11     a31 a32 ⎤⎥⎦ ⎡⎢⎣ a11 a12     a21 a22 ⎤⎥⎦         ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

There are numerous ways to obtain the inverse of a matrix using numerical methods techniques such as Gauss-Jordan elimination, Gaussian elimination, or LU decomposition. Link to numerical methods.


TRANSPOSE OF A MATRIX
The transpose of a matrix is defined by interchanging the rows and columns of the matrix. The transpose of a m × n matrix is a n × m matrix. The transpose of a matrix is usually designated by the superscript ()’ or ()T. Consider matrix A:

A=⎡⎢⎢⎢⎣ a11 a12 a13       a21 a22 a23       a31 a32 a33 ⎤⎥⎥⎥⎦

Then, A’ or AT = ⎡⎢⎢⎢⎣ a11 a21 a31       a12 a22 a32       a13 a23 a33 ⎤⎥⎥⎥⎦

Here, row one becomes column one, and the process continues until the last row becomes the last column.
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