The matrix $A$ given in the image is:

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

To determine if this matrix has an inverse, we need to calculate its determinant. If the determinant is zero, the matrix is singular, meaning it does not have an inverse. If the determinant is non-zero, the matrix is non-singular, meaning it has an inverse.

The determinant of $A$ can be calculated as follows:

$\text{det}(A) = -3 \begin{vmatrix} 2 & 3 \\ 1 & 3 \end{vmatrix} - 0 \cdot \begin{vmatrix} 0 & 3 \\ 1 & 3 \end{vmatrix} + (-5) \begin{vmatrix} 0 & 2 \\ 1 & 1 \end{vmatrix}$

Calculating each minor:

1. $\begin{vmatrix} 2 & 3 \\ 1 & 3 \end{vmatrix} = (2 \cdot 3) - (3 \cdot 1) = 6 - 3 = 3$
2. $\begin{vmatrix} 0 & 2 \\ 1 & 1 \end{vmatrix} = (0 \cdot 1) - (2 \cdot 1) = 0 - 2 = -2$

Now, substitute these into the determinant formula:

$\text{det}(A) = -3 \cdot 3 + 0 \cdot M_{12} + (-5) \cdot (-2) = -9 + 0 + 10 = 1$

Since the determinant of $A$ is 1 (which is non-zero), the matrix $A$ is non-singular and thus has an inverse. So the correct option is:

• $A$ es no singular. En otras palabras, tiene una inversa.

Would you like me to show you the steps to compute the actual inverse matrix?

Here are five related questions to consider:

1. How can we find the inverse of a 3x3 matrix manually?
2. What properties does a matrix need to have to be invertible?
3. How does changing one element of a matrix affect its invertibility?
4. What is the significance of a matrix having a determinant of zero?
5. How is matrix inversion used in solving systems of linear equations?

Tip: When computing the inverse of a matrix, always start by confirming it has a non-zero determinant to save time on calculations.