Matrix Algebra Using Determinants

In an archaeological excavation, the coordinates of three key points forming a search sector have been recorded. To calculate the area of this sector, we first need to obtain the value of the determinant of the following position matrix $A$:

$A = \begin{pmatrix} 2 & 1 & 3 \\ -1 & 0 & 4 \\ 1 & -2 & 1 \end{pmatrix}$

Calculate the value of $|A|$ using Sarrus's rule.

Calculate the inverse matrix $A^{-1}$ of the matrix $A = \begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix}$ using the adjugate matrix method:

$A^{-1} = \frac{1}{|A|} \times (\text{Adj}(A))^t$

Follow these steps:

1. Find the determinant $|A|$.

2. Calculate the matrix of cofactors $\text{Adj}(A)$.

3. Take the transpose and divide by the determinant.

Calculate the value of the determinant of the following matrix $4 \times 4$ expanding by the elements of the second row (the one with the most zeros):

$|D| = \begin{vmatrix} 1 & 3 & 0 & 5 \\ 0 & 2 & 0 & 0 \\ -1 & 1 & 4 & 2 \\ 2 & 0 & 1 & 3 \end{vmatrix}$

Without expanding the matrix completely, apply the properties of determinants to justify why the following determinant is equal to $0$:

$|A| = \begin{vmatrix} 1 & 2 & 3 \\ 5 & 10 & 15 \\ 0 & 1 & -4 \end{vmatrix}$

(Hint: Observe the relationship between the first and second rows).

Let $A$ be a square matrix of order $3$ whose determinant is $|A| = -2$. Use the properties of determinants to calculate the following values:

1. a) $|A^t|$ (determinant of the transpose).
2. b) $|A^{-1}|$ (determinant of the inverse).
3. c) $|3A|$ (determinant of the matrix multiplied by the scalar $3$).

Given that the determinant of a square matrix $A$ of order 3 is $|A| = -2$, use the properties of determinants to calculate, with reasoning, the value of the following determinants:

1. a) $|A^t \times A|$

1. b) $|3A^{-1}|$

1. c) The determinant of the matrix resulting from multiplying the first row of $A$ by $5$, swapping the second and third rows, and transposing the matrix.

A courier company uses a matrix-based encryption system. For a message to be "decryptable", the matrix used $M$ must be invertible.

Given the matrix $M = \begin{pmatrix} k & 4 \\ 1 & k \end{pmatrix}$, determine for which values of the real parameter $k$ the matrix has no inverse (that is, it is a singular matrix).

In an air traffic control system, the position of three drones at a given moment is defined by the columns of a matrix $P$. In the simplified model used, the configuration is considered safe only if the determinant of $P$ is non-zero. Given the matrix:

$$P(k) = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$$

1. a) Calculate the values of $k$ for which the drones would be in a "critical" configuration (zero determinant).

1. b) For $k = 0$, check whether the matrix is invertible and, if so, find its inverse $P^{-1}$ using the adjugate matrix method.

1. c) Show that for any value of $k \neq 1$ and $k \neq -2$, $P(k)$ is non-singular.

Given the matrix $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 1 & -1 \\ 3 & 0 & 2 \end{pmatrix}$, calculate:

1. a) The determinant of $A$ using Sarrus's rule.

1. b) The matrix of cofactors $\text{Adj}(A)$.

1. c) Check that the property $A \times \text{Adj}(A)^t = |A| \times I$ holds, where $I$ is the identity matrix of order 3.

Given the matrix $M = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$:

1. a) Determine the values of the real parameter $k$ for which the matrix $M$ is singular (has no inverse).

1. b) For the value $k = 0$, calculate the inverse matrix $M^{-1}$ using the adjugate matrix method.

Consider the matrix $A = \begin{pmatrix} 1 & 0 & -1 \\ 0 & m & 3 \\ 4 & 1 & -m \end{pmatrix}$.

1. a) Determine the values of the parameter $m$ for which the matrix $A$ has an inverse.

1. b) For $m = 2$, solve the matrix equation $X \times A - 2 \times Adj(A) = |A| \times I$, where $Adj(A)$ is the adjugate matrix of $A$ and $I$ is the identity matrix.

Hint: Use the fundamental relationship between a matrix and its adjugate: $A \times (Adj(A))^t = |A| \times I$.

Solve the following matrix equation to find the unknown matrix $X$:

$A \times X = B$

Where:

$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$

(Remember: $X = A^{-1} \times B$. You must calculate $A^{-1}$ first).

Solve the following matrix equation $A \times X \times B = C$, where the matrices $A$, $B$ and $C$ are:

$A = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 3 & 1 \\ 1 & 1 \end{pmatrix}$ and $C = \begin{pmatrix} 2 & 0 \\ -1 & 4 \end{pmatrix}$

Justify beforehand that the matrices $A$ and $B$ are invertible by calculating their determinants.

A logistics company uses an encryption algorithm based on matrix algebra to protect its routes. The security key is a matrix $X$ that satisfies the following equation:

$$A \times X \times A^t = B - 2I$$

where $I$ is the identity matrix of order 2 and the matrices $A$ and $B$ are:

$$A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 10 & 12 \\ 4 & 14 \end{pmatrix}$$

1. a) Justify, using determinants, that the matrix $A$ is invertible.

1. b) Rearrange the above equation for the matrix $X$.

1. c) Explicitly calculate the matrix $X$.

Solve the system of matrix equations to find the matrices $X$ and $Y$:

$\begin{cases} 2X + Y = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} \\ X - Y = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \end{cases}$

Once found, calculate the determinant of the product matrix $X \times Y$.

Solve the following system of matrix equations for the unknown matrices $X$ and $Y$:

$$\begin{cases} 2X + Y = A \\ X - Y = B \end{cases}$$

where the coefficient matrices are:

$$A = \begin{pmatrix} 5 & 1 \\ 11 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & -1 \\ 1 & 2 \end{pmatrix}$$

Once $X$ and $Y$ have been found:

1. a) Calculate $|X \times Y|$ and $|X + Y|$.
2. b) Determine whether the product matrix $X \times Y$ is invertible using its determinant.

It is known that a square matrix $B$ satisfies the equation $B^2 - 3B + 2I = O$, where $I$ is the identity matrix and $O$ is the zero matrix.

1. a) Show that $B$ is a non-singular matrix (has an inverse) and express $B^{-1}$ in terms of $B$ and $I$.

1. b) If $B$ is of order 2 and its determinant is $|B| = 2$, what is the determinant of the adjugate matrix of $B$, $|\text{Adj}(B)|$?

Let $A$ and $B$ be two square matrices of order 3 such that $|A| = 2$ and $|B| = -1$. Calculate, justifying each step using the properties of determinants, the value of the following determinants:

1. a) $|(A^t \times B)^{-1}|$

1. b) $|3 \times A^2 \times Adj(B)|$

1. c) Given that a matrix $C$ satisfies $C^2 = C$ (idempotent matrix), show that its determinant can only be 0 or 1.

An electronic components company manufactures three types of circuits: $C_1$, $C_2$ and $C_3$. The production cost of each circuit is determined by the determinant of the material matrix $M$:

$M = \begin{pmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{pmatrix}$

1. a) Knowing that $x$, $y$ and $z$ represent the quantities of silicon, copper and tin respectively, show that the determinant of this matrix (Vandermonde) is $(y-x)(z-x)(z-y)$.

1. b) If for a specific circuit the quantities are $x=1$, $y=2$ and $z=4$, is the matrix $M$ invertible? Justify your answer using the result from the previous section.

Given the square matrix of order $n$:

$$D_n = \begin{pmatrix} x & a & a & \dots & a \\ a & x & a & \dots & a \\ a & a & x & \dots & a \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a & a & a & \dots & x \end{pmatrix}$$

1. a) For the case $n = 3$, show that the determinant of the matrix is $|D_3| = (x+2a)(x-a)^2$.

1. b) Generalise the previous result for a matrix of order $n$ and deduce the expression for $|D_n|$ in terms of $x, a$ and $n$.

1. c) If $x = 5$ and $a = 2$, for which values of $n$ would the matrix $D_n$ be singular?

Consider the matrix of order 4 depending on the real parameter $k$:

$$M = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1-k & 1 & 1 \\ 1 & 1 & 1+k & 1 \\ k & k & k & k^2 \end{pmatrix}$$

Using only the properties of determinants, perform the following tasks:

1. a) Find the value of the determinant of $M$ in terms of $k$, giving the result in factorised form.

1. b) Determine the values of $k$ for which the matrix $M$ has no inverse.

1. c) For the case $k = 2$, calculate the value of the determinant of the adjugate matrix of $M$, that is, $|Adj(M)|$, without explicitly calculating the elements of this adjugate matrix.

Determine the rank of the following matrix $B$ by studying its minors:

$B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 2 & 4 & 6 \end{pmatrix}$

Give reasons to state whether the rank is $1$, $2$ or $3$.

Consider the matrix $A = \begin{pmatrix} 1 & m & 1 \\ m & 1 & 1 \\ 0 & 2 & 2 \end{pmatrix}$.

1. a) Calculate the determinant of $A$ as a function of the parameter $m$.

1. b) Determine the rank of the matrix $A$ according to the values of $m$.

Determine the rank of the matrix $A$ according to the values of the real parameters $a$ and $b$:

$$A = \begin{pmatrix} 1 & a & 1 & 2 \\ 2 & b & 2 & 4 \\ 1 & 1 & a & 2 \end{pmatrix}$$

To carry out the study, use the method of bordered minors and explain in detail under what conditions the rank is 1, 2 or 3.