Calculating determinants
Sarrus' rule only works for 3rd-order determinants. For a $4\times4$ matrix, or to save work on a $3\times3$ with many parameters, you need other tools: making use of zeros, breaking the determinant down into smaller "pieces" (Laplace) or turning the matrix into a triangular one before multiplying its diagonal.
Determinant of a triangular matrix
A logistics company uses an upper triangular matrix to organize its delivery routes efficiently. Calculate the determinant of the matrix $L$, knowing that for this type…
A logistics company uses an upper triangular matrix to organize its delivery routes efficiently. Calculate the determinant of the matrix $L$, knowing that for this type of matrix the calculation is greatly simplified:
$$L = \begin{pmatrix} 5 & -2 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 2 \end{pmatrix}$$Minor and cofactor
Given the matrix $M$, calculate its minor $\alpha_{21}$ and its cofactor $A_{21}$: $$M = \begin{pmatrix} 2 & 1 & 3 \\ 4 & 0 & 5 \\ -1 & 2 & 1 \end{pmatrix}$$
Given the matrix $M$, calculate its minor $\alpha_{21}$ and its cofactor $A_{21}$:
$$M = \begin{pmatrix} 2 & 1 & 3 \\ 4 & 0 & 5 \\ -1 & 2 & 1 \end{pmatrix}$$Expansion along a row or column: Laplace's method
Calculate the value of the following determinant of order 4 by expanding along the elements of the most suitable row or column (Laplace expansion): $$\Delta =…
Calculate the value of the following determinant of order 4 by expanding along the elements of the most suitable row or column (Laplace expansion):
$$\Delta = \begin{vmatrix} 2 & 0 & 3 & 4 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 5 & 0 \\ 3 & 2 & 0 & 1 \end{vmatrix}$$A GPS positioning system uses four satellites to triangulate a position in space. The volume of the tetrahedron formed by the four satellites is related to the absolute…
A GPS positioning system uses four satellites to triangulate a position in space. The volume of the tetrahedron formed by the four satellites is related to the absolute value of the determinant of their coordinates matrix. Calculate with reasoning the value of the following determinant of order 4, which represents a specific configuration of these satellites:
$$\Delta = \begin{vmatrix} a+1 & a & a & a \\ a & a+1 & a & a \\ a & a & a+1 & a \\ a & a & a & a+1 \end{vmatrix}$$Express the result as a polynomial in $a$ simplified as far as possible.
Triangularising with Gaussian elimination before calculating
Calculate the value of the determinant of the matrix $M$ of order 4 by first transforming it into an upper triangular matrix using Gaussian elimination (creating zeros…
Calculate the value of the determinant of the matrix $M$ of order 4 by first transforming it into an upper triangular matrix using Gaussian elimination (creating zeros below the main diagonal):
$$M = \begin{pmatrix} 1 & 2 & 0 & 3 \\ -1 & 0 & 2 & 1 \\ 0 & 1 & -1 & 2 \\ 2 & 3 & 1 & 0 \end{pmatrix}$$We know that the determinant of a matrix $A$ of order $2 \times 2$ is $|A| = 5$. If we multiply all the elements of the second row by $3$, what will be the value of the…
We know that the determinant of a matrix $A$ of order $2 \times 2$ is $|A| = 5$. If we multiply all the elements of the second row by $3$, what will be the value of the new determinant? Justify your answer based on the properties.