Properties of determinants

Suppose you are analysing a matrix $A$ of order $2 \times 2$ representing the profits of two branches of a clothing store in Madrid and Barcelona. The determinant of this matrix is $|A| = 12$.

Calculate, without performing the full expansion, the value of the following determinants by applying the corresponding properties:

1. a) The determinant of the transpose of the matrix: $|A^t|$.
2. b) The determinant of the matrix if we swap the Madrid row with the Barcelona row.
3. c) The determinant of the matrix if the profits for both cities were doubled (that is, calculating $|2 \cdot A|$).

Given a matrix $C$ of order 3 with a column consisting entirely of zeros:

$C = \begin{pmatrix} 4 & 0 & 1 \\ -1 & 0 & 5 \\ 9 & 0 & 2 \end{pmatrix}$

Reason, without using Sarrus' rule, why the determinant of this matrix is necessarily zero.

A graphic design application stores the coordinates of a logo in a matrix $M$ of order 3. The system informs us that the determinant of this matrix is $|M| = 0$.

Explain with reasons what you can state about the rows of the matrix $M$. Does the inverse matrix $M^{-1}$ exist? Justify your answer based on the properties of determinants.

Without calculating the value of the determinants, justify why the following determinant is equal to zero:

$\begin{vmatrix} 2 & 4 & 8 \\ 1 & 0 & 7 \\ 3 & 6 & 12 \end{vmatrix}$

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

Given the following matrix $B$:

$B = \begin{pmatrix} 1 & 5 & -2 \\ 0 & 3 & 4 \\ 1 & 5 & -2 \end{pmatrix}$

State, without needing to apply Sarrus' rule, the value of its determinant $|B|$. Mention the specific property that has allowed you to reach this conclusion immediately.

Consider two square matrices $X$ and $Y$ of order 3. It is known that $|X| = 4$ and $|Y| = -2$.

Calculate the value of the determinant of the product of both matrices, $|X \cdot Y|$, stating which property you are using.

In a strategy video game, the attack strength of two units is given by the determinant of the matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. We know that the value of this determinant is $|A| = -5$.

If the player activates a "booster" that triples only the values of the first row, what will the new value of the determinant be? Justify your answer using the property of multiplying a line by a real number.

It is known that the determinant of a matrix $A = \begin{pmatrix} x & y \\ z & t \end{pmatrix}$ is equal to $7$.

From this information, calculate the value of the determinant of the matrix $B = \begin{pmatrix} z & t \\ x & y \end{pmatrix}$. What property have you applied to know whether the sign changes or stays the same?

Let $A$ be a square matrix of order 3 whose determinant is $|A| = -2$. Using only the properties of determinants, calculate the value of the following determinants, justifying each step:

1. a) $|A^t|$, where $A^t$ is the transpose matrix of $A$.

1. b) $|3A|$.

1. c) $|A^{-1}|$, assuming that $A$ is non-singular.

1. d) The determinant of the matrix resulting from swapping the first and third rows of $A$.