Determinants of order 3 — Sarrus' rule
With a $3\times3$ matrix there is no longer a single diagonal of each kind: you have to complete three products "to the right" (positive) and three "to the left" (negative). This method is called Sarrus' rule.
Sarrus' rule
To avoid missing any product, mentally repeat the first two rows (or columns) just below the matrix: this way you can "see" the three descending diagonals and the three ascending ones without leaving the drawing.
Calculate the value of the following determinant using Sarrus's rule. Make sure you calculate the products along the main and secondary diagonals paying careful…
Calculate the value of the following determinant using Sarrus's rule. Make sure you calculate the products along the main and secondary diagonals paying careful attention to the signs:
$$ \left| \begin{array}{ccc} 1 & 2 & -1 \\ 0 & 3 & 2 \\ 4 & -1 & 1 \end{array} \right| $$Find the value of the unknown $x$ so that the determinant of the following matrix is zero: $$ \left| \begin{array}{ccc} x & 0 & 2 \\ 1 & 1 & 3 \\ 0 & 2 & 1 \end{array}…
Find the value of the unknown $x$ so that the determinant of the following matrix is zero:
$$ \left| \begin{array}{ccc} x & 0 & 2 \\ 1 & 1 & 3 \\ 0 & 2 & 1 \end{array} \right| = 0 $$Using Sarrus to decide whether a matrix has an inverse
The same criterion as for order 2 still holds for order 3: if $|A| = 0$ the matrix is singular (it has no inverse); if $|A| \ne 0$ it is regular (non-singular). With three rows, "collapsing" means that the three rows lie in the same plane instead of filling all three dimensions.
In a triangular matrix (upper or lower), the determinant is simply the product of the elements on its main diagonal. Calculate the determinant of the following matrix…
In a triangular matrix (upper or lower), the determinant is simply the product of the elements on its main diagonal. Calculate the determinant of the following matrix representing a company's production costs:
$$ \left| \begin{array}{ccc} 5 & 8 & 12 \\ 0 & 2 & 7 \\ 0 & 0 & 3 \end{array} \right| $$A mobile app development company is analysing the impact of three of its star features (A, B and C) in three different markets. The performance data are collected in the…
A mobile app development company is analysing the impact of three of its star features (A, B and C) in three different markets. The performance data are collected in the following matrix $M$:
$M = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 3 \\ 2 & 1 & 1 \end{pmatrix}$
Calculate the determinant of the matrix $M$ using the Sarrus rule (write the three products of the positive diagonals and the three of the negative ones) to determine whether the analysis model is consistent (that is, whether its determinant is non-zero).
Area of a triangle and volume of a parallelepiped
The same idea extends to three dimensions: the volume of a parallelepiped defined by three vectors $\vec{u}, \vec{v}, \vec{w}$ starting from the same vertex is the absolute value of the determinant of the matrix whose rows are their components. If that determinant is $0$, the three vectors are coplanar: they lie in the same plane and do not "fill" any volume.
Calculate the determinant of the following matrix, which represents the number of three types of shares (Technology, Energy, Banking) in three different investment…
Calculate the determinant of the following matrix, which represents the number of three types of shares (Technology, Energy, Banking) in three different investment portfolios:
$$ \left| \begin{array}{ccc} 10 & 10 & 10 \\ 2 & 3 & 5 \\ 1 & 0 & 2 \end{array} \right| $$(Hint: Try taking a common factor out of the first row before applying Sarrus to simplify the numbers).
For a digital security system to be valid, the "signature matrix" $S$ must not be singular (its determinant must be non-zero). Given the matrix:…
For a digital security system to be valid, the "signature matrix" $S$ must not be singular (its determinant must be non-zero). Given the matrix:
$S = \begin{pmatrix} 1 & a & 0 \\ 2 & 1 & 1 \\ a & 0 & -1 \end{pmatrix}$
Determine the values of the real parameter $a$ for which the security system would not be valid.
In the design of a new map for a strategy video game, the vertices of a protected triangular region are given by the coordinates of the points $P(1, 1)$, $Q(3, 5)$ and…
In the design of a new map for a strategy video game, the vertices of a protected triangular region are given by the coordinates of the points $P(1, 1)$, $Q(3, 5)$ and $R(k, 7)$. Point $R$ can be anywhere on the line $y = 7$, depending on the value of $k$. The game engine requires the area of this region to be exactly $10$ square units in order to balance the factions.
Use the area formula with order-3 determinants to find the possible values of the parameter $k$ that satisfy this condition.