Determinants of order 2
To every square matrix we can assign a single number that sums up very useful information about it: whether it has an inverse, whether its rows are "independent", and what area it generates. For $2\times2$ matrices, this number is found with a very simple calculation.
The determinant of a 2×2 matrix
The product of the leading diagonal (solid) is added; the product of the other diagonal (dashed) is subtracted. The determinant is written $|A|$ or $\det(A)$, and the matrix can also be drawn between vertical bars instead of brackets.
Calculate the value of the following 2nd-order determinants, paying special attention to the use of negative signs: a) $\begin{vmatrix} 5 & -3 \\ 2 & 4 \end{vmatrix}$ b)…
Calculate the value of the following 2nd-order determinants, paying special attention to the use of negative signs:
- a) $\begin{vmatrix} 5 & -3 \\ 2 & 4 \end{vmatrix}$
- b) $\begin{vmatrix} -1 & -6 \\ 0 & 7 \end{vmatrix}$
- c) $\begin{vmatrix} 10 & 15 \\ 2 & 3 \end{vmatrix}$
Find the value of the determinant of the matrix $A$ when it contains simple fractions. This type of calculation is common when working with allocation coefficients in…
Find the value of the determinant of the matrix $A$ when it contains simple fractions. This type of calculation is common when working with allocation coefficients in economics problems:
$A = \begin{pmatrix} \frac{1}{2} & 4 \\ \frac{1}{4} & 6 \end{pmatrix}$
Determinants with an unknown
A determinant can also contain a letter instead of a number. By setting the result equal to a given value, you get an equation, usually linear or quadratic, which you solve as usual.
Determine the value of the unknown $x$ so that the following equality holds, where the result of the determinant is a known integer value:…
Determine the value of the unknown $x$ so that the following equality holds, where the result of the determinant is a known integer value:
$\begin{vmatrix} x & 5 \\ 2 & 8 \end{vmatrix} = 14$
Consider the matrix $A = \begin{pmatrix} x+1 & 2 \\ x & x-1 \end{pmatrix}$. Determine the real values of $x$ for which the determinant of the matrix is equal to $6$.
Consider the matrix $A = \begin{pmatrix} x+1 & 2 \\ x & x-1 \end{pmatrix}$. Determine the real values of $x$ for which the determinant of the matrix is equal to $6$.
Singular matrices and regular matrices
Look at the diagram: if the determinant is not $0$, the two rows of $A$ generate a parallelogram with a non-zero area; if it is $0$, the two rows are proportional and the parallelogram is "flattened" into a line, with no area. That is why a singular matrix has no inverse: there is no way to "undo" a transformation that collapses the plane onto a line.
Given the matrix depending on the real parameter $m$: $$A(m) = \begin{pmatrix} m^2 - 9 & m + 3 \\ m - 3 & 2 \end{pmatrix}$$ a) Calculate the value of its determinant in…
Given the matrix depending on the real parameter $m$:
$$A(m) = \begin{pmatrix} m^2 - 9 & m + 3 \\ m - 3 & 2 \end{pmatrix}$$- a) Calculate the value of its determinant in terms of $m$.
- b) Determine all values of $m$ for which the matrix does not have an inverse (singular matrix).
- c) If $m = 0$, calculate the determinant of the inverse matrix $A^{-1}$ using the properties of determinants, without finding the inverse matrix explicitly.
In a matrix encryption system, a key matrix $K$ of order 2 is used. For the system to be secure and allow decryption, the matrix $K$ must be invertible. The following…
In a matrix encryption system, a key matrix $K$ of order 2 is used. For the system to be secure and allow decryption, the matrix $K$ must be invertible. The following matrix is proposed:
$$K = \begin{pmatrix} k-2 & -3 \\ -2 & k \end{pmatrix}$$- a) Calculate the values of $k$ that make the system insecure (that is, such that $K$ has no inverse).
- b) If it is known that the determinant of the key matrix must be equal to $10$ to comply with a specific security standard, calculate the possible values of $k$.
The determinant as the area of a parallelogram
A determinant of order 2 can represent the area of a parallelogram defined by two vectors in the plane. Calculate the absolute value of the determinant formed by the…
A determinant of order 2 can represent the area of a parallelogram defined by two vectors in the plane. Calculate the absolute value of the determinant formed by the vectors $\vec{u} = (4, 1)$ and $\vec{v} = (2, 5)$ to find that area:
$Área = \left| \begin{vmatrix} 4 & 2 \\ 1 & 5 \end{vmatrix} \right|$
In designing a logo for a new tech startup, a designer uses a parallelogram defined by the vectors $\vec{u} = (a, 1)$ and $\vec{v} = (2, a)$. It is known that the area…
In designing a logo for a new tech startup, a designer uses a parallelogram defined by the vectors $\vec{u} = (a, 1)$ and $\vec{v} = (2, a)$. It is known that the area of this parallelogram is given by the modulus of the determinant of the matrix formed by both vectors.
Calculate the value of the parameter $a$ so that the area of the logo is $3$ square units.