Gauss's method for calculating determinants
In 1st Bachillerato you calculated determinants of order 2 and 3 using Sarrus's rule. That rule does not exist for order 4 or higher. Gauss's method solves the problem: it turns any matrix into a triangular one using operations that do not change the determinant (or that change it in a controlled way), and then all you have to do is multiply the diagonal.
The determinant of a triangular matrix
A small video game shop tracks its sales of two titles ($V_1$ and $V_2$) during the first two days of the week. The data are organised in the following matrix $A$:…
A small video game shop tracks its sales of two titles ($V_1$ and $V_2$) during the first two days of the week. The data are organised in the following matrix $A$:
$A = \begin{pmatrix} 12 & 5 \\ 8 & 4 \end{pmatrix}$
Calculate the determinant of the matrix $A$ to check whether the sales vectors are linearly independent.
Given the following matrix $M$, perform the elementary row operation $F_2 = F_2 - 2F_1$ to obtain a zero in element $a_{21}$ and write down the new resulting matrix.…
Given the following matrix $M$, perform the elementary row operation $F_2 = F_2 - 2F_1$ to obtain a zero in element $a_{21}$ and write down the new resulting matrix. Does the value of the determinant change when performing this operation?
$M = \begin{pmatrix} 1 & 3 \\ 2 & 10 \end{pmatrix}$
What each operation does to the determinant
A logistics company organises its routes using a cost matrix $M$. For the system to be efficient, the determinant of this matrix must not be zero. Apply row operations…
A logistics company organises its routes using a cost matrix $M$. For the system to be efficient, the determinant of this matrix must not be zero. Apply row operations (Gaussian elimination) to determine whether the following matrix is non-singular (its determinant is non-zero):
$$ M = \begin{pmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 3 & 7 & 11 \end{pmatrix} $$What linear dependence relationship do you observe between its rows after carrying out the first step of Gaussian elimination?
Triangularising step by step
Calculate the value of the following determinant of order 4 by first transforming it into an upper triangular matrix using Gaussian elimination (making zeros below the…
Calculate the value of the following determinant of order 4 by first transforming it into an upper triangular matrix using Gaussian elimination (making zeros below the main diagonal):
$$ |A| = \begin{vmatrix} 1 & 2 & -1 & 3 \\ 2 & 5 & 0 & 4 \\ -1 & -2 & 3 & -2 \\ 3 & 6 & -3 & 10 \end{vmatrix} $$Consider the matrix of order 4 depending on a real parameter $k$:…
Consider the matrix of order 4 depending on a real parameter $k$:
$M = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1+k & 1 & 1 \\ 1 & 1 & 1+k^2 & 1 \\ 1 & 1 & 1 & 1+k^3 \end{pmatrix}$
Use Gaussian elimination (triangulation) to transform the matrix and calculate its determinant in terms of $k$. Subsequently, discuss for which values of $k$ the matrix does not have an inverse.
Gauss's method with parameters
When the matrix depends on a letter, triangularising with Gauss's method is usually more convenient than Sarrus or Laplace: the zeros appear as functions of the parameter and the determinant comes out factorised naturally, which makes it easier to discuss for which values it vanishes.
Find the rank of the following matrix $A$ depending on the values of the real parameters $a$ and $b$, using Gaussian elimination:…
Find the rank of the following matrix $A$ depending on the values of the real parameters $a$ and $b$, using Gaussian elimination:
$A = \begin{pmatrix} 1 & a & 1 & 2 \\ 2 & b & 2 & 4 \\ 1 & a & a & 2 \end{pmatrix}$
Describe in detail all possible situations for the rank of the matrix depending on the relationships between $a$ and $b$.