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The problem asks to find the inverse of the matrix \( T \), where:

\[
T = \begin{pmatrix}
-1 & 2 & -3 \\
7 & 3 & 0 \\
4 & 2 & 5
\end{pmatrix}
\]

### Steps to Find the Inverse of a 3x3 Matrix

Given a matrix \( T \), its inverse \( T^{-1} \) can be found using the following formula:

\[
T^{-1} = \frac{1}{\text{det}(T)} \cdot \text{adj}(T)
\]

Where:
- \(\text{det}(T)\) is the determinant of \( T \).
- \(\text{adj}(T)\) is the adjugate (or adjoint) of \( T \), which is the transpose of the cofactor matrix.

#### Step 1: Find the Determinant of \( T \)

\[
\text{det}(T) = -1 \cdot \begin{vmatrix} 3 & 0 \\ 2 & 5 \end{vmatrix} - 2 \cdot \begin{vmatrix} 7 & 0 \\ 4 & 5 \end{vmatrix} + (-3) \cdot \begin{vmatrix} 7 & 3 \\ 4 & 2 \end{vmatrix}
\]

Calculating the 2x2 determinants:

\[
\begin{vmatrix} 3 & 0 \\ 2 & 5 \end{vmatrix} = 3(5) - 0(2) = 15
\]

\[
\begin{vmatrix} 7 & 0 \\ 4 & 5 \end{vmatrix} = 7(5) - 0(4) = 35
\]

\[
\begin{vmatrix} 7 & 3 \\ 4 & 2 \end{vmatrix} = 7(2) - 3(4) = 14 - 12 = 2
\]

So,

\[
\text{det}(T) = -1(15) - 2(35) + (-3)(2) = -15 - 70 - 6 = -91
\]

#### Step 2: Find the Adjugate of \( T \)

The cofactor matrix of \( T \) is obtained by finding the minors and applying the sign pattern:

\[
\text{Cofactor Matrix} = \begin{pmatrix}
\begin{vmatrix} 3 & 0 \\ 2 & 5 \end{vmatrix} & -\begin{vmatrix} 7 & 0 \\ 4 & 5 \end{vmatrix} & \begin{vmatrix} 7 & 3 \\ 4 & 2 \end{vmatrix} \\
-\begin{vmatrix} 2 & -3 \\ 2 & 5 \end{vmatrix} & \begin{vmatrix} -1 & -3 \\ 4 & 5 \end{vmatrix} & -\begin{vm

Find the inverse matrix of T where T = [[-1, 2, -3], [7, 3, 0], [4, 2, 5]]

Learn how to calculate the inverse of a 3x3 matrix T = [[-1, 2, -3], [7, 3, 0], [4, 2, 5]] by using determinants and adjugates. This detailed guide walks through the steps of finding the determinant, the cofactor matrix, and the adjugate to determine the inverse matrix. Ideal for students at the undergraduate level studying linear algebra.

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