Properties of determinants

Knowing that the determinant of a square matrix $A$ of order 3 is $|A| = -4$, use the properties of determinants to calculate the value of the following determinants:

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

1. b) The determinant of the matrix resulting from swapping the first two rows of $A$.

1. c) The determinant of the matrix obtained by multiplying the third column of $A$ by $-5$.

Without performing any operations or applying Sarrus' rule, justify why the following determinants are zero:

1. a) $\begin{vmatrix} 2 & 1 & 3 \\ 4 & 2 & 6 \\ 1 & -1 & 5 \end{vmatrix}$

1. b) $\begin{vmatrix} 1 & 5 & 0 \\ 2 & 3 & 0 \\ -4 & 7 & 0 \end{vmatrix}$

1. c) $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 5 & 7 & 9 \end{vmatrix}$

Prove, using the properties of determinants, that if a linear combination of the other rows is added to a row of a square matrix, the value of the determinant does not change. To do so, check what happens to the determinant:

$|A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$

if we perform the row operation $F_2 = F_2 + k \cdot F_1$.

Let $M$ be a matrix of order $3 \times 3$ such that its determinant is $|M| = 2$. Calculate the value of the determinant of the matrix $3M$. Explain which property you have used and take care with the order of the matrix.

In a strategy video game, the coordinates of resources are stored in a matrix $3 \times 3$ called $C$. If we know that $|C| = 10$, what would be the value of the determinant if we double all the rows of the matrix (that is, we calculate $|2C|$)? Compare this result with the one obtained if we only doubled the first row.

Given two square matrices $A$ and $B$ of the same order, we know that $|A| = 3$ and $|B| = -2$. Calculate, stating the properties applied:

1. a) $|A \cdot B|$

1. b) $|B^2|$

1. c) $|A \cdot B^t|$

If $A$ is a matrix of order 3 such that $|A| = \frac{1}{2}$, calculate the determinant of its inverse matrix $A^{-1}$, justifying your answer using the property of the determinant of the product of matrices.

Let $A$ be a square matrix of order 3 such that its determinant is $|A| = -2$. Using exclusively the properties of determinants, calculate the value of the following determinants:

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

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

1. c) $|\text{det}(A) \cdot A|$