To determine if the matrix $A$ has an inverse, we first calculate its determinant. If the determinant is non-zero, the inverse exists; otherwise, it does not.

Given: $A = \begin{bmatrix} 1 & -1 & -1 \\ 0 & -2 & -1 \\ -3 & -4 & 0 \end{bmatrix}$

Step 1: Compute the Determinant of $A$

The determinant of a $3 \times 3$ matrix $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ is calculated as: $\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

For $A = \begin{bmatrix} 1 & -1 & -1 \\ 0 & -2 & -1 \\ -3 & -4 & 0 \end{bmatrix}$:

• $a = 1$, $b = -1$, $c = -1$
• $d = 0$, $e = -2$, $f = -1$
• $g = -3$, $h = -4$, $i = 0$

Substitute these values: $\text{det}(A) = 1((-2)(0) - (-1)(-4)) - (-1)((0)(0) - (-1)(-3)) + (-1)((0)(-4) - (-2)(-3))$ $= 1(0 - 4) + 1(0 - 3) - 1(0 - 6)$ $= 1(-4) + 1(-3) - 1(-6)$ $= -4 - 3 + 6 = -1$

Since the determinant is $-1$ (non-zero), the matrix $A$ has an inverse.

Step 2: Calculate the Inverse of $A$

The inverse of a $3 \times 3$ matrix $A$ can be found using the formula: $A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$ where $\text{adj}(A)$ is the adjugate of $A$.

Adjugate of $A$

The adjugate is the transpose of the cofactor matrix of $A$.

I'll proceed to compute this and present the inverse matrix to you.The inverse of matrix $A$ is:

$A^{-1} = \begin{bmatrix} 4 & -4 & 1 \\ -3 & 3 & -1 \\ 6 & -7 & 2 \end{bmatrix}$

This matrix provides the unique inverse, given that the determinant of $A$ is non-zero.

Would you like additional details on any part of the solution, or do you have further questions?

Here are five related questions that might help deepen your understanding:

1. How do you find the determinant of larger matrices, such as $4 \times 4$?
2. What properties make a matrix invertible or non-invertible?
3. How can you verify that two matrices are inverses of each other?
4. What role does the adjugate matrix play in calculating the inverse?
5. How does matrix inversion apply in solving systems of linear equations?

Tip: For matrices larger than $3 \times 3$, determinant calculations can become lengthy, and computational tools are often used for efficiency.