A matrix can be used to store a translation (position). In computer graphics, 3x3 matrices are used to store the scaling and rotational information, but if another row is added, for example, if we create a 3x4 or have a 4x4 matrix, the last row can literally store the X, Y, and Z positional information.
Example 4.2. Compute the eigenvalues of the matrix B from example 4.1 and assign the values to a vector b. We do this by typing the following: >> b = eig(B) b = 1 8 3 2. The eigenvalues are 1, 8, 3, 2. There are four of them because our matrix is 4x4. Notice also that it is very easy to compute the determinant of B. All we have to do is
By definition the determinant here is going to be equal to a times d minus b times c, or c times b, either way. ad minus bc. That's the determinant right there. Now what if we were to multiply one of these rows by a scalar. Let's say we multiply it by k. So we have the situation a, b.
1 Answer. Let's look for the Smith Canonical Form, since it is your goal behind the question. For t = 1 t = 1, we get in particular x − 1 x − 1 and x + 1 x + 1, whose gcd g c d is 1 1, so the gcd of all 1 × 1 1 × 1 minors is 1 1. For t = 2 t = 2, we get in particular (x − 1)2 ( x − 1) 2 and (x + 1)2 ( x + 1) 2, whose gcd g c d is 1 1
A determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. Here is the source code of the C++ Program to Compute Determinant of a Matrix. The C++ program is successfully compiled and run on a Linux system.

Every time I reduced this to row echelon form, I got $\dfrac{1}{48}$ as the determinant when the actual determinant is $48$. Here are the row operations. The rows that I have highlighted are the ones that change the determinant since we are changing a row by a factor. All the other operations don't change the determinant and we never switch

NmvF. 276 109 476 148 335 405 128 372 12

determinant of a 4x4 matrix example